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Edge influence on the response of shear layers toacoustic forcingRienstra, S.W.
DOI:10.6100/IR40127
Published: 01/01/1979
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Citation for published version (APA):Rienstra, S. W. (1979). Edge influence on the response of shear layers to acoustic forcing Eindhoven:Technische Hogeschool Eindhoven DOI: 10.6100/IR40127
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EDGE INFLUENCE ON THE RESPONSE OF SHEAR LA YERS
TO ACOUSTIC FORCING
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE
HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. DR. P. VAN DER LEEDEN, VOOR
EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP
VRIJDAG 8 JUNI 1979 TE 16.00 UUR
DOOR
SJOERD WILLEM RIENSTRA
GEBOREN TE MANADO (INDONESIË)
1979
DRUKKERIJ J.H. PASMANS, 'S-GRAVENHAGE
Dit proefschrift is goedgekeurd
door de promotoren
Prof.dr.ir. G. Vossers
en
Prof.dr. D. G. Crighton
T.1 ]. 2 1. 3
2
2. 1 2.2 2.3 2.4 2.5 2.6
3
3. 1 3.2 3.3 3.4 3,5 3.6 3. 7 T8-
4
4. 1 4.2 4.3 4.4
4.5 4.6 4. 7
CONTENTS
Contents
Abstract
Introduction Aeroacoustics The problems aonsidered The Kutta condition in aaoustia problems
The diffraction of a cylindrical pulse and of a harmonie plane wave at the trailing edge of a semi-infinite plate submerged in a uniform flow Introduction Formulation of the problems The pulse from a source The harmonie plane wave Comparison with experiments Conclusions
The interaction of a harmonie plane wave with the trailing edge, and its subsequent vortex sheet, of a semi-infinite plate with a uniform low-subsonic flow on one side Introduction Causality vs Kutta condition Formulation of the problem Formal solution Approximation of the split funations Approximation for small Mach number Comparison with experiments Ccï::: Z~ci:;ns
The interaction between a subsonic uniform jet flow, issuing from a semi-infinite circular pipe, and a harmonie plane wave with small Strouhal number Introduction Formulation of the problem Formal solution Approximations, for small Strouhal number, for the axial velocity and pressure inside the flow, and for the far field Extensions downstream Comparison with experiments Conclusions
5 Acoustic waves interacting with viscous flow near an edge: application of results from the triple deck theory
page 1
2
4 8
12
15 1 7 19 23 24 26
27 30 32 34 44 48 55 55
57 60 63 70
83 86 87
5.1 Introduction 89 5.2 Formulation of the problems 93 5.3 Harmonie plane waves with a wake 95 5.4 Harmonie plane waves with a mixing layer 97 5.5 Conclusions 99
6 Appendices 6. J Propertiés of Fresnel 's -integral F ó.2 Properties of Lerch's transcendent ~
References
Levensloop
Nawoord
Samenvatting
100 102
104
110
110
2
ABSTRACT
This thesis considers the interaction of sound with a flow, Three models
are studied. In all the models a central element is the presence of a
trailing edge. A uniform subsonic inviscid compressible flow on either or
only one side of a semi-infinite thin flat plate, and in a semi-infinite
thin-walled open tube is perturbed by sound.
In the problem with a·flow on both sides of the plate, a cylindrical sound
pulse and a plane harmonie sound wave are scattered by the edge and the
flow. Explicit results, exact within the usual linearisation, are obtained
for the resulting field. The level of singularity of the solution,imposed
at the edge, determines the amount of vorticity shed from the edge. These
vortices make themselves felt acoustically in the diffracted wave. Upstream,
the pressure amplitude of the regular solution (maximum vortex shedding;
the Kutta condition is valid) is lower than the one of the singular
solution without vortex shedding; downstream it is higher.
In the problem with a flow on one side of the plate, a plane harmonie sound
wave, coming from inside the flow upstream, is scattered again at the edge
and generates vorticity, depending on the imposed edge singularity. However,
now we have a tangential velocity discontinuity (vortex sheet) past the
plate, which is unstable in the Helmholtz sense. The vortices, if shed,
will trigger this instability and modify the hydrodynamic field considerably.
Exact solutions in integral form are given, where it is shown that it is the
plate edge condition that governs the instability, rather than a causality
condition, introduced by previous authors. From the exact sclutions explicit
approximations are obtained, valid for small values of the Mach number.
In the tube problem, plane harmonie sound waves come from inside the pipe.
Beside diffraction and vortex shedding, reflection at the open pipe exit is
present as well. Exact solutions, again in integral form, are approximated
by explicit expressions in the limit of long sound waves (small Strouhal
number). Independent of the generated vortices, the "end correction" in the
case of an incompressible flow appears to differ essentially from the one
in the case of no flow, (viz 0,2554 and .0,6133 respectively). This is a
consequence of the non-uniform double limit "Mach number -> O, Strouhal
number -> O".
Virtually all of the induced acoustic energy is reflected at the pipe exit
in the case of the singular solution without vortex shedding; in the case
of the regular solution an amount of acoustic energy of order M (Mach
number) is transmitted. However, this is transformed into hydrodynamic
energy (vertices), and therefore hardly felt in the far field.
3
Finally, the first two roedels are extended to include viscosity to give
some rational basis for the vertex shedding at the edge. Known results from
the "triple deck" theories for incompressible flow are adapted. For large
values of the Reynolds number, the Kutta condition is shown to be
appropriate to leading order in laminar flow, and with the values of the
parameters restricted to a certain range; furthermore, the correction term,
due to viscosity, is given.
4
chapter 1
INTRODUCTION
1.1 Aeroaaoustias
In this thesis we shall discuss some problems which can be classified
as belonging to, or related to, aeroacoustics.
"_ àetine aeroacoustics as the intermediate discipline between fluid
Hlechanics är:.d acoustics, describing the product ion or- -;UUJ.iil by ~! ~" ~"n
the interaction of sound with flow. The usual definition (see for example
Goldstein (1976)) only involves thto :0"nd production. Of course, from the
practical point of view it is only the net radiated sound that is
important, while in some theoretical problems the interaction can b0
neglected (or rather, taken for granted). When the sound is produced by
the flow itself and then interacts with the flow, we will inevitably have,
in the stationary state, a closed loop without beginning or end, and it is
sufficient to take the saturated flow pattern as given and then calculate
the resulting sound field. It is, however, also clear that the interaction
aspect is fundamental anyway. And in systems with an external sound source,
where the interest is in the dependence of the radiated sound on the source
properties we cannot avoid the interaction. Therefore, we include it in
our definition.
Looking back in history, we can recognize several of the classical
acoustic problems (e.g. the aeolian harp, the flute, the organ-pipe, the
sound-sensitive flame) to be basically of aeroacoustic nature. At the time,
however, the fluid dynamical aspects were considered and explained only
ad hoc, and the problems were not seen to have any binding element but the
acoustics. Before their coherence was understood, several related problems
had to be solved. As often in scientific developments, the driving force
here was a technical need, namely the noise of airplanes and turbo
machinery.
Before 1950, moving air was not felt to be an important source of
noise. The available engine power was indeed too low. As a consequence,
5
the published theories on this subject were very rare (the best known
being Gutin's propeller theory in 1937, see Goldstein (1976)). A dramatic
turn carne around 1950 when the first commercial jet engine-driven airplanes
were introduced (not altogether "on the quiet", one might say!).
Considering the notorious noise of those first jet transports, it can
hardly be coincidence (nor indeed was it) that the history of the
understanding of aerodynamic noise really started in 1952. At that time
Lighthill (1952, 1954) reformulated the compressible, time dependent,
three dimensional Navier-Stokes equations without external farces, into
a form reminiscent of the equation for acoustic waves, viz.
( 1. 1. 1)
where p is the density, c0
a constant reference sound speed,xi and t the
space and time coordinates, and T .. = pv.v. +p .. - c 2 p6 .. is Lighthill's lJ 1 J lJ 0 lJ
stress tensor. In Tij we have the velocity vi in the xi-direction, 6ij the
Kronecker delta function, and the compressive stress tensor pij' In most
atmospneci~ and underwater applications pij is given by
1 3 pij = p6ij - 2µ(eij - 3 6ij L: ekk), where p is the pressure, µ the
k=l
coefficient of viscosity and e .. the rate of strain tensor, lJ
e .. lJ
Equation (1. 1. 1) is exact, equivalent to the Navier-Stokes equations
and not preferable in particular when we deal with an arbitrary fluid
mechanical problem. It is, however, most preferable in problems concerning
the noise produced by a flow, and radiated into a still ambient medium.
When we have a flow of bounded extent, embedded in a still medium (still
in the sense that here the characteristic variations in the terms of the
tensor Tij are much smaller than in the flow), and we take for c0
the
mean value of the sound speed c in the still medium, then equation (1.1.1)~
6
applied in the still medium, just describes the sound radiated by the flow
into the still medium in a very natura! way.
Although exact, equation (1. 1. 1) is equivalent, for an observer in the
still medium, to the inhomogeneous acoustic wave equation. So provided we
know T .. in the flow, we can calculate the consequent outside sound field 1J
in the classica! way. This is called "Lighthill 's acoustic analogy". This
point of view on the Navier-Stokes equations has been shown to provide in
a remarkably elegant way much insight in the kind of problems just mentioned.
Of course, one can argue that, strictly speaking, to know Tij is only
possible if we solve the complete problem and therefore, owing to the
complexity of the Navier-Stokes equations, in practice possible only in
very simplified examples. However, Lightbill's formulation, aided by the
nature of the human ear, is so well suited to the problem of sound
produced by flow that only crude knowledge of Tij may well suffice. The
human ear is hardly sensitive to differences in sound pressure but
sensitive rather to the logarithm of it. As a consequence, estimates based
on similarities and sealing laws do in many practical cases provide enough
information on T .. to calculate the radiated sound power with fair accuracy. 1J
In particular, turbulent flow, so poorly known in detail, is relatively
well described by simple sealing laws, and therefore most susceptible to
treatment by the analogy.
Take, for example, homogeneous isotropic turbulence, which we can
model as composed of small uncorrelated eddies. When the typical
fluctuation velocities, say of order U, are low-subsonic, so that
uc- 1 << 1 (though this is not necessarily true of the mean flow), the
eddies are acoustically compact and the right hand side of (1. 1. 1) can be
identified with a quadrupole source distribution. Taking advantage of that,
Lighthill derived his celebrated P - U8 law for the total radiated sound
power P. Following essentially the same lines, Curle (1955), Ffowcs
Williams & Hawkings (1969) and Ffowcs Williams & Hall (1970), to name but
a few, extended the theory to include the presence of surfaces. Ffowcs
Williams (1963) showed how to treat acoustically non-compact eddies; here,
although the fluctuation Mach number uc- 1 is assumed to remain small, the
eddies are rendered spatially non-compact by virtue of their convection by
the mean flow at a supersonic Mach number. Much more has been done: the
influence of thermodynamic inhomogeneities has been studied, Lighthill 's
equation (1. 1. 1) has been generalized for non-uniform mean flow, but we
shall not consider these aspects further. They all exploit the acoustic
analogy for turbulent flow, and that is just not the theme of the present
studies in the chapters to carne.
Although very successful in many cases, Lighthill's approach seemed
to fail in some applications where the details of the flow are important.
We shall study two of these cases. When near the trailing edge of an
airfoil, in the presence of flow, sound is generated either by turbulent
eddies or by an external source, the sound may induce vorticity shedding
from the edge which dominates, or at least provides a local ordering of
7
the turbulent eddies. As a result, the assumption of homogeneous turbulence
is no more correct. This effect may be negligible, but certainly not
always. The edge region contributes relatively much to the total radiated
sound because of the acoustically efficient sound radiation properties of
turbulent eddies near an edge (Ffowcs Williams & Hall (1970)), and because
moving vorticity generates sound near an edge (Crighton (1972 c)).
The other problem relates to the noise from a real full-scale jet
engine. Although the noise from model-scale laboratory jets follow
Lighthill's predicted U8 -law very well for moderate Mach numbers, substantial
deviations are found at almost all conditions in the operation of jet
engines (Crighton (1975)). Same of the explanations proposed for this
"excess noise" problem are that vorticity shed from the edge (Crighton
(1972 b)) and large coherent structures (big eddies) play a role (Moore
(1977)), The eddies arise either from the vorticity shed from the edge by
external acous tic or hydrodynamic forcing, or spontaneously from "nowhere",
possibly generated by their own turbulent sound, while their growth is a
result of the unstable character of a mixing-layer. The noise is specifically
produced by the interaction of the shed vertices with the edge (Crighton
(1972 b)) and by the vortex pairing process of the big eddies downstream
(Ffowcs Williams & Kempton (1978)). In bath the airfoil and the jet
problems, a lot of additional, as yet almost unknown, information is
necessary before we can describe Lighthill's tensor Tij' and thus the
radiated sound field, properly.
Obviously, a deterministic theory is most preferable for the
description of these relatively orderly mechanisms.
8
However, a theory capable of predicting the sound produced by eddies
born of shed vorticity does not exist.yet. The development of turbulence
is hardly understood and virtually impossible to describe. The only
feasible ways to attack this problem seem to involve careful measurements
(Moore (1977)), or computer simulations (Acton (1976)), and then the
calculation from this improved information on T .. of the resulting sound l.J
field (cf. Ffowcs Williams & Kempton (1978)).
On the other hand, as far as the sound is concerned, that is generated
directly at the edge by the vorticity shed from the edge, we have better
prospects for a deterministic theory. This is especially true with the
presence of an external sound source of sufficient intensity and coherence
to shed vortices large compared with the turbulent eddies. An inviscid
uniform mean flow with small disturbances is then likely to be the
appropriate model. This has the extremely important advantage that exact
solutions are not unlikely to be achieved anymore.
This is basically the approach which we shall follow to treat the
prohlems mentioned. Of course, we could have posed the formalized problems
ad hoc, as a mathematica! problem of "convected ·diffràcl:ion''. HN~Cvèr, ~
felt it necessary to place them firmly in the context of modern aero
acoustics. Although aeroacoustics has its emphasis traditionally on
turbulence, we believe that the "clean" sorts of problem, like those
which we will treat here, will supply necessary complementary knowledge in
all those cases in which the structure of the turbulent flow is modified
by the sound itself. A more detailed description of the problems to be
treated will be given in the next section, 1.2.
1.2 ~bZems aonsidered
In the following chapters we shall dlscuss some problems concerning
the interaction of ~ flow with diffracting sound waves. ln ch2pters 2 and 3
we have a semi-infinite rigid flat plate in an inviscid compressible.medium;
This medium flows along the plate, in chapter 2 on both sides and in
chapter 3 on one side only, with a uniform subsonic velocity directed from
and parallel ~o the plate. In this setting we consider the related problems
of a cylindrical pulse (chapter 2), and a harmonie plane wave (chapters 2
and 3) diffr~cting at the trailing edge. In chapter 4 we have a semi-
infinite rigid cylindrical pipe, issuing a subsonic inviscid compressible
jet flow into the ambient medium, the latter being at rest relative to the
pipe. A harmonie plane wave, incident from inside the pipe, reflects and
diffracts at the pipe exit.
In all these problems viscosity plays a centra! rele, but only at one
point, the trailing edge. Therefore, we can, to a certain extent, neglect
the explicit viscous aspects and make the problems, which are impossible to
solve in genera! otherwise, analytically tractable. However, in view of the
centra! rele of viscosity, any theory, however limited, that includes it
is worthwhile to study. In an attempt to meet this we shall in chapter 5
briefly present an extension of the calculations of chapters 2 and 3 to
include viscosity. The present knowledge on this subject encompasses no
more than rather particular, possibly unrealistic, problems, but it is
believed that these show the genera! mechanism at least in principle.
9
We shall explain why the viscous farces at the edge are important to
the whole field. It is well-known that, without the presence of flow, the
field of a sound wave dif fracting at a sharp edge of a rigid plate contains
an r-!-singularity in the velocity at the edge. When we have on one or both
sides of the plate a flow, while the edge is a trailing edge, this
singularity is often (but not always) smoothed by the flow by means of the
shedding of vorticity from the edge. These shed vertices change th 0 total
sound field, because vorticity moving near a solid edge generates sound
(Crighton (1972 c)). They also may change the hydrodynamic flow field,
when they trigger instabilities, as in our chapters 3 and 4. Therefore it
would be worthwhile to include this smoothing process, resulting in the
vertex shedding, in acoustic-flow interaction problems. However, this
smoothing process is essentially a viscous effect and therefore not
included in an inviscid model, and hence the me del adopted should be a
viscous one. Yet an inviscid model is much preferable, because the
presence of viscosity makes things usually unfeasibly compiicated.
Fortunately an inviscid model with mean flow is rich enough to incorporate
the vertex shedding at will. Only the amount of shed vorticity, or,
equivalently, the level of singularity of the velocity at the edge, is
undetermined then, and must be included as a boundary condition of the
problem. There is, however, not simply one particular condition correct for
all occurring cases, as we will see. A whole variety is possible and we
hardly know when they apply. The best we can do, therefore, is to consider
10
two relevant extremes, and compare them to quantify the effect of the shed
vortices.
This boundary condition on the edge singularity has to follow either
from a solution of the viscous equations, or from the interpretation of
suitable experiments. For example, ·we may conclude from observations that
the shed vorticity results in a smooth flow at the edge, and consequently
require the field in the model to be suitably smooth at the edge, the
so-called "Kutta condition". It is also possible, in other experiments,
that we conclude that no vorticity is shed at alL Then the additional
boundary condition in- the inviscid model has to be that the total amount
of vorticity in a finite volume surrounding the edge is the same in both
the steady and the perturbed case. We note that in an explicit solution
the term representing the shed vorticity is usually easily recognized and
can simply be dropped to satisfy this last condition, Obviously there may
be intermediate cases where vorticity shedding does take place, but not
enough to make the Kutta condition fully valid. More will be said about
this mechanism of vortexshedding in the next section, 1.3.
Calculations involving viscosity (see chapter 5) tend to support the
validity of the Kutta condition in laminar flow. Also in clean and well
controlled experiments the Kutta condition usually seems to apply
(Heavens (1978) , Bechert & Pfizenmaier (1975)), although the intermediate
and no-vortex-shedding states are easily achieved too (Archibald (1975) ,
Bechert & Pfizenmaier (1975) , Heavens (1978)), Parameters controlling
these differences are no doubt the dimensionless viscosity, frequency and
amplitude, but there are probably more. The precise dependence is not
known, especially in turbulent flow, and a genera! rule cannot be given.
Only in the problem of chapter 4, the subsonic jet perturbed by sound
waves, is there experimental evidence that the Kutta condition is always
to be applied when the viscous boundary layer nèar the nozzle exit is
thin and the sound wavelength is large, both with respect to the pipe
radius (Bechert & Pfizenmaier (1975)),
It is clear that studying a model with one particular prescribed
edge condition is not very useful if we do not know when this condition is
valid. On the other hand, in some cases the contribution of the shed
vortices to the sound field may be not important at all, and then there is
nothing to worry about. Therefore we shall calculate in chapters 2 and 3
both the Kutta condition solution and the no-vortex-shedding solution,
Il
being probably the physically relevant extremes, with the other intermediate
solutions in between. In chapter 4 we shall present in detail the Kutta
condition solution, this being here the most likely one, and we shall
present a few characteristics of the no-vortex-shedding solution for
comparison. The application of the Kutta condition, compared with the case
without shed vorticity, introduces highly significant differences to the
diffracted wave in the wake problem of chapter 2, and to practically the
whole sound field in the pipe problem of chapter 4. In the low-subsonic
plane mixing layer problem of chapter 3(essentially, of course, the high
frequency limit of the pipe problem) only the hydrodynamic field is
affected considerably, rather than the sound field.
A remarkable feature, encountered in the harmonie wave/mixing layer
problems of chapters 3 and 4, is the question of causality. In the
existing literature the myth has grown up that some causality condition
is necessary to achieve the correct solution. We shall show, however, that
this is a misunderstanding due to improper modelling.
Let us consider some aspects of the models. In the first place we
have the assumption of harmonie sound waves, i.e. waves behaving
sinusoidally in time for all time. Secondly, in earlier models the solid
body (plate, pipe) was not included in order to simplify the problem
further. As a first step, these assumptions look reasonable. However, a
simplification restricts the validity of a model. A very important question
is, therefore, "to which physical problems do these most simplified models
apply?". The answer is simply: "none". To understand this we have to
investigate the real character of these simplifications. Essentially they
consist in the assumptions that the trailing edge is very far away, and
that the start of the source is very long ago, both taken in the limit to
infinity. It appears that this double-limit is not uniform, i.e. the two
limits may not be interchanged. Now consider for convenience a harmonie
line source. In limit (i), when first the start of the source is taken
back in time and then the body withdrawn in space, the field becomes
infinite everywhere. The instability field, growing exponentially in space,
is triggered by the vorticity, shed from the edge by the only-algebraically
decaying incident field, and fills the whole space. In the other case,
12
limit (ii), when first the body is withdrawn and then the switch-on time
of the source decreased to minus infinity, the sound field never reaches
the edge, so no vorticity is shed from the edge and the instability
encountered is only the transient response of the mixing layer. Because
of this prominent transient dependence the solution resulting from this
last limit was called "the causal solution".
Obviously, in every physical setting, inevitably including an edge
from which the shear layer develops, this causal solution cannot be what
is normally meant by a "harmonie solution". So the only relevant limit is
limit (i), although unfortunately yielding a singular solution. However,
this was not previously realised. The general opinion was, that only
limit (ii) gives the relevant solution because it is the only limit that
gives ~ solution. Even more, when the models were extended to include
solid bodies like flat plates and pipes, the authors concerned, misled by
the fermer results, insisted on causality (i.e. that the field is from a
source, switched on a long time ago), although the whole instability field
is now ruled by the vertex shedding edge, and not by the start of the
source. The transient response field is swept away, because it cannot match
to the edge, and the whole notion of causality is superfluous. The only
condition that matters is the level of singularity (i.e. the amount of, shed
vorticity) at the edge. Because of its central role in this thesis, it is
worthwile to devote section 1.3 to it, and there we try to explain in
intuitive terms the responsible mechanism.
1.3 The Kutta condition in aaoustia problems
Consider the problem of a harmonie sound wave diffracting at the
edge of a flat plate, without flow to start with. A close look at the
solution shows that the pressure has a square root behaviour at the edge,
and is positive on one side of the plate and negative on the other,
alternating in time. This results in an alternating flow negotiating the
edge, and therefore to inverse square root velocity singularities at the
edge. When the edge is a trailing edge of a uniform inviscid flow on one
or both sides of the plate, these acoustic velocity perturbations around
the edge are usually too small to deflect the main flow, and the alternating
pressure differences across the plate now result in alternating velocity
perturbations tangential to the plate and discontinuous across the
streamline emanating from the edge. (This is regardless, of course, of
any possible discontinuity in the main flow velocity). A tangential
velocity discontinuity is equivalent to vorticity distributed along a
surface, and therefore we call the formation of the tangential velocity
discontinui ty "vortex shedding". We note that this vort ex shedding is very
unlikely in the case of a leading edge. The shed vortices are then greatly
hindered by the plate,
So in trailing edge problems with acoustic perturbations we have to
admit vortex shedding, However, when we do not knowhow much vorticity is
shed, the solution is not uniquely determined, so the level of vortex
shedding must be added as an extra boundary condition (see section 1.2).
We may have,for instance, so much vortex shedding that the pressure (which
itself is singular in the presence of mean flow and absence of vortex
shedding) has no singularity at the edge any more. The corresponding
boundary condition of a smooth pressure is called the "Kutta condition".
The smoothing process, responsible for the vortex shedding, is essentially
a viscous effect, so we have to seek for viscous models if we want a
better understanding of it.
13
The understanding of t~e mechanisms involved has lately been
considerably deepened by the discovery of the asymptotic laminar boundary
layer structure near a flat plate trailing edge (the triple deck) by
Stewartson (1969) and Messiter (1970), and the application of it in several
related problems by Stewartson and co-workers (Riley & Stewartson (1969),
Brown & Stewartson (1970), Brown & Daniels (1975),Daniels (1978)). In
chapter 5 we shall give a few explicit applications to our acoustic
problems.
We have, for laminar flow at high Reynolds number, the following
picture. Due to the change in boundary conditions the flow in the boundary
layer accelerates when it passes the edge. This gives rise to a singularity
in the pressure of the inviscid outer flow, and this singularity is
smoothed in a small region around the edge. However local this flow induced
pressure may look, it is nonetheless essential for the transition from the
wall boundary layer to the wake or mixing layer. lt is shown in the papers
mentioned above that the Kutta condition problem can be identified with
the balance between that flow induced pressure and the externally generated
14
pressure perturbations (i.e. diffracting sound waves in our case).
When the pressure of the Kutta condition solution is of the same order
of magnitude (near the edge) as the flow induced pressure, the viscous
srnoothing forces, prepared for the flow induced pressure singularity, take
care of both and the Kutta condition is valid. Probably this is also the
case when the externally generated pressure is much lower (Daniels (1978)),
although Bechert & Pfizenrnaier (1975) argue that for very small amplitudes
the diffracting sound wave effectively only sees stagnant flow close to the
edge where its pressure singularity becomes important. Hence, for decreasing
pressure amplitude, the Kutta condition rnay then eventually not be valid.
Finally, when the external pressure dominates the flow induced pressure,
separation is likely to occur and the Kutta condition is not valid. With
respect to this last point it should be noted that the sound pressure
singularity rnay be helped to overcome the srnoothing forces by another
external mechanism capable of generating a singularity at the edge. Exarnples
rnay be a plate at incidence, or a·wedge shaped edge. The two singularities
add up, and the Kutta condition is then violated earlier than if only one
external singularity-inducing process were in action.
We go on now in t~e next three chapters to present analyses of the
outer problems with vortex shedding, and then return in chapter 5 to the
viscous aspects of these flows.
chapter 2
.THE DIFFRACTION OF A CYLINDRICAL PULSE AND OF A
HARMONIC PLANE WAVE AT THE TRAILING EDGE OF A
SEMI-INFINITE PLATE SUBMERGED IN A UNIFORM FLOW
~ Introduotion
In their analysis of the r8le of a sharp edge in aerodynamic sound,
Ffowcs Williams & Hall (1970) did not consider the effect of mean flow,
and thus the possibility of vertex shedding, which would smooth the
acoustic singularity at the edge (see section 1,3). There is no reason
15
to expect vertex shedding when we deal with a leading edge, hut, as we saw
in section 1.3, in the case of a trailing edge the Kutta condition is often
satisfied.
Now to investigate the consequences of the application of this
condition, Jones (1972) calculated exactly (in the acoustic approximation)
the fields, with and without the Kutta condition, of what is probably the
simplest relevant model of edge-flow-sound interaction. He considered
in a uniform inviscid flow a harmonie line source parallel to a semi
infinite plate of which the edge is a trailing edge. The vertices, if
shed, are assumed to be convected along.the wake. To model this
the tangential velocities are allowed to be discontinuous across the thin
planar wake, Jones compared the continuous solution (without shed
vorticity and singular at the edge) with the Kutta condition solution (with
shed vorticity and smooth at the edge), and concluded that the field only
changed considerably in the neighbourhood of the wake, In the Kutta
condition solution a non-decaying wave along the surface of the wake is
present.
edge,
Of course, this is just the convected vorticity shed from the
An important aspect is, that this surface wave only exists in the
velocity, and disappears in the pressure, Jones noted this, hut
unfortunately did not pay much attention to it. He did not give the
16
pressure explicitly, and it was not at all clear how pressure measurements
would decide whether the continuous or the Kutta condition solution is
relevant in a particular case. Indeed, this insufficiency of information
was a source of confusion in the interpretation of experiments by Heavens
(1978).
Heavens made schlieren photographs of the diffraction of a sound
pulse at an airfoil trailing edge in a subsonic flow. He observed that
only the diffracted wave amplitude varied considerably with the prevailing
flow conditions.
strongly visible;
In unsteady or separated flow the diffracted wave was
in a smooth flow it was weak. Heavens suggested that
the Kutta condition was connected with this difference, and concluded with
the opinion that his observations tend to support the conclusion of Howe
(1976), that trailing edge flows are quieter if they do not show singular
behaviour, in contrast to, for instance, the predictions of Jones' (1972)
model.
If this were true it would have been very surprising. In Howe's
model the vanishing of the diffracted wave is only found if the source is
moving with the mainstream velocity and the Mach number is low. Jones'
model is more suitable in regard to Heavens' experiments and should have
better prospects. The only problem is that schlieren photographs
essentially measure density (and so pressure) gradients, and this makes
Jones' model not immediately applicable, as we saw in the above.
In this chapter we shall calculate the diffracted pressure wave of a
pulse generated by a line source, and the complete timeharmonic field of
incident plane waves. The pulse solution shows us the validity of Jones'
model and yields a good explanation of Heavens' findings. Although there
is in a linear problem no essential difference between the experimentally
relevant information provided by the pulse-and the harmonie solution, here
the pulse solution is more transparent and is preferable. The plane wave
solution shows an alternative way to find Jones' eigensolution (here by
inspection, in contrast to Jones' original rather complicated calculations
involving the Wiener-Hopf technique), and it may be useful for possible
future experiments, as the amplitude of a plane wave is more readily
measurable. The plane wave solution itself was already found by Candel
(1973), but he seems not to have realised that he had the eigensolution,
representing the convected vorticity, at his disposal.
Our principal conclusions are that application of the Kutta condition
results in an increase of the diffracted pressure in a downstream are,
symmetrically surrounding the wake, and a decrease elsewhere. In
particular in the pulse problem, only the pulse part of the diffracted
pressure is affected and not the tail,
We finally mention the application of Jones' (1972) eigensolution in
17
a slightly different context. Davis (1975) studied the sound radiated by
the vortices shed from an airfoil with a blunted trailing edge. He
modelled these vortices by a vortexsheet in the wake surf ace, and calcu-
lated the field in the low Mach number limit. However, he <lid not seem
to realise that he applies this limit. Moreover, he tried to satisfy the
Kutta condition by adding a field which is claimed to be an eigensolution
but in fact is not, as Howe (1976) noted. It is readily verified that
the corresponding total acoustic energy in a volume surrounding the wake
is infinite, and this is only possible if energy is brought into the
system via the surface of the wake. Nevertheless, Davis' correct
solution - which does not satisfy the Kutta condition - is just the low
Machnumber approximation of Jones' (1972) eigensolution, and we will see
in the subsequent sections that the schlieren photographs, made by
Lawrence, Schmidt & Looschen (1951) and reproduced by Davis (1975) to
illustrate his theory, once more confirm Jones' model.
2.2 Formulation of the problems
As we model the problems in two-dimensional form, we introduce the
coordinates (x,y) = (r cos8, r sin8), where - n < 8 ~ n Now consider
the following two (closely related) problems,
together by figure 1.
A sketch of both is given
18
main flow
={>
plate
I
IShadow boundary / without convection
I / diftracted wave __ ~~ boundary
_ -- -- -- with convection
wake ___ z-axis
-... __ shadow boundary ..._-.... ~ith convection
\ --, \
Figu:r>e 1
\ shadow boundary \without convection
\
A uniform inviscid subsonic flow parallel to the x-axis with
dimensionless velocit.Y (l ,O), density 1 and sound speed M-l , surrounding
a semi-infinite flat plate located along the negative half of the x-axis
(y = O, x < O; or: e = rr), is perturbed for t ~ 0 by a pulse from a
line source at (x0
, y0
) = (r0
cose0
, r0
sine0
) in the upper half-space
(i.e. e0
E (O,rr)) , and by a harmonie plane wave propagating from below
in the direction ei , 0 < ei < rr • The perturbation quantities are
(u, v) and p for velocity and pressure respectively. The amplitudes
are considered to be small enough to justify the sound speed being taken
as constant and the perturbation of the wake position being negligible.
The regions containing vorticity are consequently restricted to lines and
points, and we can intróduce a potential ~ , defined by V~ = (u,v).
Linearizing, we have the following equations and boundary conditions:
32 (-ax2
p
x < 0
x > 0
- (2.... + 2....) "' at ax '!' ,
ay <!> (x,O)
0 in the harmonie case ,
0
<a"t + a") [q,(x, + 0) - q,(x, - 0)·1 0 (no pressure diff erence across the wake) ,
19
(2. 2 .1) a
ay q,(x, + o) 1-qi(x -0) 'iJy • (the wake is a
material surf ace)
t < 0 q, (x,y,t) p(x,y,t) 0 (dropped in the harmonie case)
The harmonie field is radiating outwards at infinity.
Both the pulse - and the harmonie field tend to zero far
from the plate and the wake.
2.3 The pulse from a source
We anticipate properties of <!> (as a generalised function),
sufficient to guarantee the existence of its Fourier transform
$ (x,y,w) f00
<l>(x,y,t) exp (-iwt) dt •
After transformation of the equations (2.2.1), we have a problem for $ which is almost the same as the one solved by Jones (1972). The dif fer-
ence is that Jones assumed a priori the wake to have a particular form.
However, it can be shown that this form is necessary and would result in
any case. Therefore, we can take for $ simply Jones' solution. We
split $ into the continous part $c and an eigensolution $e
where A is determined by the Kutta condition applied to $ . Following
20
Jones we def ine Il e 1
1. 1 1+13 og M , x = Il X = Il R cose ,
y = R sine (where - 11 < e ~ 11), x0
Il X0
=Il R0
cos80
, y0
= R0
sin80
(similar to the classical Prandtl-Glauert transformation), and introduce:
F(z) = exp(i z2 ) Joo
exp(-it2 ) dt (a version of Fresnel's integral), z
G(x,y,w) = exp (-i w Mr 1
cosh u) du + éosh u) du
(the usual Green's function for the semi-infinite rigid plate diffraction
problem), where
r 2 = r 2 + r 2 - 2r r cos(e+e ) (the distance from the source and its
1,2 0 0 0
image respectively) with the obvious modification to R1
, 2 , and
± arsinh {
!
(rr )2
e+2eo} 2--0 - cos
rl,2
The index 1(2) belongs to the upper(lower) sign of t and ± As
usual the symbol H will denote the Heaviside stepfunction.
$ (x,y,w) e
exp(i Î) 213/iî
r { ! e-201} { , l_F (2 wSM R)' sin + F (2 wllM R) 2
exp [- M wM2 i ~ R + i - (X-X ) ] + s s 0 .
- J__l-exp(- i wMR cos(8+0 )) + exp(- i wMR cos(0-0 ))] 213 . s 1 Il 1 •
H (-0)
A
1 sin 1e ( 2118 )
2 2 0 2 wMR
0 cos~0 1
exp(-i wM R - i 11) s 0 4 .
Then we have:
(2. 3.1)
The transform of <Pc back into the time domain is most conveniently
perf ormed by using a well known solution of the rigid plate dif fraction
problem without flow, We have
<Pc(x,y,t)
where T 0
1 211
ij!(X,y,T ) 0
St + M2 (X-X ) 0
2~ f00
G(X,y,w')exp(iw'T0
)dw'
and ij! is the solution of
The function ij! is given by Friedlander (1958,p.126)(with the plate lying
along the right half of the x-axis, and with another source strength),
according to which <Pc is given by
<P (x,y, t) c
H(T0
-MR1
)
(T~-M2Ri) 2
+ H(T
0-MR
2) 1
(T2-M2R2) 2 0 2
8-8 {l+sgn(cosT) H (MR+MR -T ) } +
0 0
8+8 {1-sgn(cos --0
) 2
H(MR+MR -T )} 0 0
(2.3.2)
Consequently the diffracted pressure is, ignoring terms not containing
o-functions, given by
p d(x,y,t) c,
where T
l-Mcos8
cos8+cos8 0
(2.3.3) (RR ) 2
0
It turns out that the pressure pe corresponding to ~e has a very
simple form and is most suitable for transformation into the time domain:
A pe (x,y,w) sinj8sinj8 l wM2
-2 o exp i -0
- (X-X ) - iwM (R+R ) e2 (RR l, ,_, o e o
0
(2.3.4)
(Section 6.2, an appendix concerning the function F, may be of help for
the derivation.) An:lso the eigensolution in the time domain, which
21
22
together with the continuous solution .(2,3,3) satisfies the Kutta condition,
is
p k(x,y,t) e, A p (x,y,w) exp(iwt)dw
e
sin!e sin!e - 2 ° ,f(T)
e2 (RRo)' (2.3,5)
Note the complete agreement in time dependence of Pe,k and the pulse-
part of Pc,d • Summing these two gives
o (T) l+Mcose sinjesin!e 0
- 2 0
Pk,d Pc,d + Pe,k S2M cose+cose (RR ) '
0 0
(2.3.6)
Clearly we have
Pk,d l+M co se 0 1 when + 0 < 0 < 11 - 0 > - 11
Pc,d 1-M cos0 0 0
i.e. - 11 + e < e < 11 - e 0 0
< 1 elsewhere
l (2.3. 7)
So when we compare the smooth solution with the continuous solution at a
fixed observation point, we see from (2.3,7) that application of the Kutta
condition results in an increase of the pressure amplitude within the
symmetrie downstream wedge lel < 11 - e0
and a decrease elsewhere. Note
that this wedge angle is only dependent upon the angle e0
of the source
location, and not on the Mach number. Of course the increase and decrease
is only of Pk,d relative to pc,d' and the actual difference may be small.
Due to the sin!e term the predicted difference will be largest near the
plate, where sin!e ~ ±1 , and small near the wake, where
This is, moreover, enhanced by the convection: note that
sinje ~ o. 1 sin!e I < 1 sinl9 I
when O < lel < Î, and lsin!el > lsin!el when }- < lel < 11 As will
become clear in section 2.5, where a comparison with experiments is made,
it is not difficult to verify the upstream decrease experimentally, hut to
check the downstream increase well-chosen Mach numbers and locations of the
source are necessary.
23
2.4 The ha1'monia plane wave
Equations (2.2.1) are valid again, but now without the source term.
Instead we have a plane harmonie wave incident from below in the direction
Above the plate there is a shadow region (es,n) , Because
of convection we have e. f 6 1 s
In fact the following relation is valid
(introduce for convenience the auxiliary variable e ) : s
The basic incident wave is then
S sine s
cose - M s
q, (x,y,t) 0 .
exp [i w~2 X-i wSM (X coses + y sines) + iwt]
using the same notation as in section 2.3. Applying again the Prandtl-
Glauert transformation in the classical Sommerfeld screen diffraction
.solution (Jones (1964)) we get the following continuous solution (which
does not yet satisfy the Kutta condition)
4> (x,y,t) c,s
TI M wM2 /IT exp [ i 4 - i wS R + i -
13- X + iw t _] .
[
! 8-8 wM ! 8 +es l • F {(2 wt R)
2
sin-~}+ F {(2 T R) sin-2-}
(2.4.1)
(index s of <jl denotes "Sommerfeld"), and the corresponding pressure c,s
p (x,y,t) c,s
- iw(l-M 132
cos8 )<jl (x,y,t) --1-(2wM) s c,s s!TI s
!
• exp ( i TI4 i wM R + i wM2 ) S - 13 - X + iwt
with the aid of section 6,1.
sin je cos!es T
(2.4.2)
Since both <l>c,s and pc,s satisfy the governing equations, so must
the remaining term in (2.4.2). Moreover, this term vanishes as r +
and bas an integrable singularity in r = 0 ,
24
soit must be a multiple of the eigensolution (2.3.4). Indeed, as $ c,s
is regular as r + 0 , pc,s becomes regular ~nd so satisfies the Kutta
condition) when we cancel that term. So we have, in a very simple way,
the solution that satisfies the Kutta condition in the form
pk (x,y,t) ,s iw
(1 - M cos El ) $ (x,y,t) 62 s c,s
(2.4.3)
and, after integration of the pressure equation of (2.2.1) (use (6.1.13)),
and application of the boundary conditions at y = 0 and infinity,
$k (x,y,t) ,s
! WM2 A $ (x,y,t) -2(2M(l-M)) cos~El exp(i-;;-X +iWt}$ (x,y,w).
c,s s P o e
(2.4.4)
Solution ~c,s can be obtained from $c by taking R0
+ 00 in a proper
way, and similarly we can find the expressions for the diffracted pulse
corresponding to pc,s
reveals nothing new.
and
2.5 Comparison with experiments
from and This, however,
As far as we could trace, the most relevant experiments for the
present theory are those by Heavens (1978). He made schlieren photographs
of the density field (which is up to a factor equal to the pressure field)
around an airfoil trailing edge. Under varying flow conditions a plane
pulse was generated above the trailing edge. This pulse was reflected by
the airfoil, transmitted through the wake, and it produced a cylindrical
diffracted wave at the edge. In addition to the pulses, also harmonie
waves were applied, but Heavens drew no conclusions from the results
because of the poor quality of the pictures.
Heavens' principal results are that under the different flow
conditions only the diffracted wave changes; it is weak in smooth flows,
where the Kutta condition may be expected to apply, and it is strong in
unsteady and separated flows, where the Kutta condition is probably not
valid. (As is mentioned in section 1.3, two different external effects
may work together and suppress the viscous farces and so invalidate the
Kutta condition.) See for instance Heavens' figure 7(a) and (b).
If we take into account the fact that the pictures do not really show
an increase or decrease in the region surrounding the wake, which
would be consistent with the small sin!G factor our conclusions,
presented in and below equation (2.3.7), may very well be the explanation
25
of Heavens' findings. As far as the upstream decrease is concerned, this
is well confirmed, while the downstream increase is not refuted. Even
more, the fact that confirmation of the downstream increase would be mor
difficult is part of the conclusions. From equations (2.3.3), (2.3.5)
and (2.3.7) we can see the effect of variations of the parameters in
possible future additional experiments complementing Heavens' series of
tests. A Mach number close to (but smaller than) one enlarges the relarive
increase near the wake, but also reduces the factor sinjB fora fixed 8,
On the other hand, placing the source near the plate increases the
diffracted wave amplitude but reduces the region around the wake where the
relative increase occurs. Probably it will depend on the sensitivity of
the schlieren system how the optimal choice of the parameters is made that
reveals the behaviour of the diffracted wave near the wake.
Other experiments, worthwhile to mention, are those by Lawrencc,
Schmidt and Looschen (1951). They made schlieren photographs of ~he
density-(pressure) sound field generated by the vortexstreet behind the
blunted trailing edge of an airfoil. (Probably more experiments of this
kind have been done in the past, but these have the advantage that the
relevant pictures are reproduced by Davis (1975) and therefore readily
available.) Although these experiments do not really correspond to our
analysis, they show the appropriateness of Jones' (1972) model and his
eigensolution (2.3.4). When we, following Ddvis (1975), model the vortex
street as a vortex sheet and do not apply the Kutta condition (otherwise,
without external force, the only solution would be the trivial one), the
relevant eigensolution is Jones' eigensolution (2.3.4), of course with a
frequency and amplitude depending on the Reynolds number and StrO<Jhal
number of the real problem. The photographs show indeed the pattern,
given by (2.3.4), namely cylindrical co~vected wavefronts coming from the
trailing edge (according to (x-t)2 + y2 = (t/M) 2) with a large
26
amplitude near the airfoil and a small amplitude near the wake (according
to sinl0). Once more it is shown that it is useful to have the
eigensolution available in pressure form, rather than only as potential.
2.6 ConaZusions
The diffraction of a sound pulse and a harmonie planè wave around a
flat plate trailing edge in a moving medium has been studied. The
relevant eigensolution, already calculated by Jones (1972), is simply
obtained f rom the plane wave solution by inspection. This eigensolution
has the same source as the diffracted wave (i.e •. the trailing edge).
Therefore it will affect only, and be in phase with, .the diffracted wave.
In particular, in the pulse problem it affects only the pulse part (and
not the tail) of the diffracted pressure. Comparison of the Kutta
condition solution with the continuous solution shows an increase of the
diffracted wave in a downstream are and a decrease elsewhere. The rate
of the increase and decrease depends on the Machnumber, hut the angle of
the are does not. As far as the upstream decrease is concerned this is
very well confirmed by Heavens (1978). Although his pictures do not
contradict the downstream increase (as the wave itself is weak there) a
few additional experiments with well chosen Mach numbers and source
locations would be welcome to complement his series of tests.
chapter 3
THE INTERACTION OF A HARMONIC PLANE WAVE WITH THE
TRAILING EDGE, AND ITS SUBSEQUENT VORTEX SHEET,
OF A SEMI-INFINITE PLATE WITH A UNIFORM LOW-SUBSONIC
FLOW ON ONE SIDE
3,1 Introduction
27
Probably the simplestmodel problem with practical importance of
acoustic waves in a non-uniform medium involves an inviscid fluid with a
thin mixing layer, modelled as a vortex sheet, which séparates a uniform
flow from a stagnant flow, perturbed by acoustic waves. As far as the
acoustics are concerned, the first correct analyses of this problem were
given by Miles (1957) and Ribner (1957) for plane harmonie incident waves,
while Gottlieb (1960) extended their work for a harmonie line source. As
was demonstrated by Helmholtz in 1868 (see Batchelor (1967)) for
incompressible flow and by Landau (1944) for compressible flow, an infinite
plane vortex sheet is unstable, bath in the temporal and the spatial sense.
It seems that in every geometry involving an infinite vortex sheet there
is always one instability which can be identified with this particular
instability (actually a pair, if we include the decaying, that is growing
for negative time, complementary one). Therefore we shall refer to this
as the Helmholtz instability. Note that the expressions: instability,
instability wave, instability field, etc., will be used synonymously.
The amplitude of the Helmholtz instability wave decays exponentially with
distance from the vortex sheet, Therefore it does not contribute to the
sound field, and could be ignored by the previous authors without affecting
their main conclusions, In the initial value problem, however, as treated
for instance by Friedland & Pierce (1969) (for a line source) and Howe
(1970) ~or a point source), these instabilities must be included to obtain
a causal solution (i.e. one with no disturbances before the source is
switched on), and could not be ignored, Unfortunately, some parts of the
instability field are more singular than, say, a Dirac delta function and
28
caused some interpretational problems. Only by Jones & Morgan (1972) was
this singularity classified as an ultradistribution - loosely speaking a
distribution with complex argument - and by them the pulse problem was
solved correctly. Having then a grasp of this problem, Jones & Morgan
were able in the same paper to find the harmonie field that wo~ld arise
from a harmonie source switched on a long, finite time ago. They called
this harmonie solution "the causa! solution". In it the Helmholtz
instability wave is present with a uniquely defined amplitude, and this
solved the riddle of the undetermined instability amplitude for this
harmonie problem; not for the genera! one. For example, the problem that
should model a real physical_harmonic source/mixing layer interaction is
not yet solved by Jones & Morgan's causa! solution, because in reality the
memory of the system cannot be very long with the inherent variety of
additional perturbations. But still the experiments (Poldervaart et al
(1974)) show a perfect reproducibility, so another mechanism must be
responsible for the choice of the eigensolution (i.e. Helmholtz instability)
amplitude.
As we will show in the following sections, the mathematically
sufficient condition, which is also physically much more acceptable than
causality, is the choice of the level of singularity at the trailing edge:
"the Kutta condition", or otherwise. In the problem discussed so far no
solid boundary with a trailing edge was included, but this is obviously
only a simplification of the reality. Of course the edge may be far
enough away in some problems for us to neglect it, as in some initialvalue
problems, but in the harmonie problem all of the vertex sheet is perturbed
and the influence of the edge is not, a priori, small. In fact, in our
linear model it becomes larger as the edge is taken further away.
Our conclusion is that the doubly-infinite vertex sheet model with
harmonie perturbations is too much simplified, resulting in a basically
undeterminable Helmholtz instability amplitude. Therefore, we shall
consider in this chapter only the model of a semi-infinite vortex sheet
downstream a flat plate with harmonie disturbances. Extending the ideas
of Jones & Morgan (1972) the semi-infinite vortex sheet problem was solved
by Crighton & Léppington (1974) and Morgan (1974). Although their
solutions are correct, in our opinion, they insisted in the derivation of
the harmonie solution on the application of the causality criterion and
on}y in the last resort on the Kutta condition. They did not realise that
the Kutta condition was sufficient to ensure uniqueness, although Crighton
& Leppington found that the Kutta condition virtually implied causality,
To clarify this controversial subject in particular, we shall devote the
next section (3,2) to it, In section 3,3 we shall formulate the problem
considered in this chapter. It is a variant of Crighton & Leppington's
and Morgan's problem. Instead of a pointsource we have plane waves, with
fronts perpendicular to the plate, coming from upstream infinity in the
subsonic flow. A formal solution will be derived in section 3.4.
Although this solution is essentially already given by Crighton &
Leppington and Morgan, we shall still repeat the calculations to show
explicitly how the causality criterion can be neglected. Finally, in the
29
last sections we shall calculate the near and far field in the low Mach
number limit. This has not been dorre before, although it was proposed by
Crighton & Leppington (1974). In the case treated here, the incident wave
is simple enough to enable an explicit approximation to be derived. Use
is made of an elegant technique, proposed by Boersma (1978), to evaluate
a complex integral approximately. It is explained in section 3.5
separately, to ease the use of it again in chapter 4. As far as the
acoustics are concerned the approximation is uniform from the near field
to the far field, except for the bow wave which together with the
instability wave is only calculated in the near field. Their f ar field
would require much additional calculation, of which the worth is dubious
since the vortex sheet does not model a mixing layer very well in that
region.
We finally have to note that the case of a supersonic flow seems to be
very different, An attempt to include this is not given here. Morgan
(1974) found that in the relevant eigensolution the vortex sheet is
discontinuous at the edge, and that the vortex sheet in the least singular,
vortex shedding, solution still leaves the edge with a finite slope (and
so with discontinuous derivative). This is not observed in experiments.
As with the l//r edge singularity of the subsonic eigensolution, the
discontinuity will probably be due to the improper linearisation. An
explanation for the finite slope may be that we do not recognise Morgan's
findings either because the slope is very small, or because it is not yet
achieved experimentally. The vortices we see in the existing experiments
are shed (according to this conjecture) in the subsonic region of the
boundary layer and deceive our observations of a satisfied Kutta condition.
30
3.2 CausaZity vs Kutta aondition
Probably the strongest tool to attack the kind of problems we are
dealing with here - harmonie waves in an infinite medium - is Fourier
transformation in space coordinates. If the transformed problem can be
solved, we obtain the solution of the original problem by transformation
of the transformed solution back into the space domain. Although this
method has proved to be extremely successful in many cases, we should
never forget that it essentially sifts the subclass of Fourier transform
able solutions from the complete class of acceptable (or rather: expected)
solutions of our problem.
possible solutions.
Therefore it is arguable that we miss certain
One way to cure this is to include all kinds of
classes of generalised functions in the class of functions in which we
seek the transformed solution, so as to extend that subclass of Fourier
transformable solutions. But this is only successful if we can guess and
specify the properties of the required solution, and if we are able to
handle these generalised functions properly. This last point is not
fundamental of course, but may play an important r8le in practice.
Another way to solve the problem, if we know the behaviour of the solution,
is to prepare it, before applying Fourier transforms, by subtracting the
known singular part from the solution, so that we are left with a neat
problem. This is done by Orszag & Crow· (1970) and Crighton (1972a) to
find the (Helmholtz unstable) eigensolution of the semi-infinite vortex
sheet problem with incompressible and compressible subsonic flow
respectively, and by Bechert & Michel (1975) for several related
problems.
A third way to find the solution, especially useful in the
problems we are treating concerning Helmholtz instabilities, which are, as
growing exponentials, specifically not normally Fourier-transformable,
is to make use of the following property of the Helmholtz instability· (not
proved by us but observed in several cases). The growth rate is more or
less linearly dependent on the frequency (in the plane sheet case: exactly
linearly), and we can interchange (temporarily) spatial growth with
temporal growth by extending the frequency space to the complex plane via
values corresponding to outgoing waves. Once having a spatially stable
problem we can safely apply Fo,·~;ertransforms, and afterwards the spatially
unstable solution is found by analytic continuation in the frequency of
31
the stable solution to real frequencies.
When we apply the causality criterion, we come across this lastmethod
rather naturally. This does not imply, however, that our solution should
be causal. The method is only one of the devices to obtain the solution,
and combined with the Kutta, or an analogous, condition, they all realise
the ~ unique solution. Moreover, it would lead to contradictions, if
we were to insist on causality, irrespective of the prescribed level of
singularity at the edge. Causality requires regularity of the solution
in some halfplane of the complex frequency space (Jones & Morgan (1974)).
In a viscous flow the level of singularity at the trailing edge is governed
by viscous farces and a function of frequency and amplitude (among others).
Examples are given in chapter 5. The corresponding amplitude of the
singular eigensolution is therefore dependent on the frequency, but
obviously not necessarily in an analytic way. It would be very artif icial
to expect the viscous farces to take into account the start of the acoustic
source very long ago.
Finally, we note the following argument to explain why the Kutta, or
an analogous, condition virtually implies causality. According to a
theorem, found by Bechert & Michel (1975) for incompressible flow and by
Howe (1976) for compressible flow, the semi-infinite vortex sheet problem
has only one eigensolution, namely a Helmholtz instability which does not
satisfy the Kutta condition. It can be interpreted as the result of
vorticity shed from the edge. In the cases considered, vorticity can only
be shed by external means. When a sound wave passes the edge without
shedding vertices (the Kutta condition is then as a consequence not
satisfied), we get the stable solution without exponential growth of the
vortex sheet displacement amplitude. The more vorticity is shed the more
of the eigensolution must be included, until finally the Kutta condition is
satisfied. Therefore the whole field is determined by the source via the
shed vorticity, and is thus causal, in the sense that the source causes the
disturbances. When in particular the Kutta condition is applied the above-
mentioned theorem precludes perturbations (eigensolutions) without a
source and now the solution is also causal in the strict sense of Jones &
Morgan (Morgan (1974)). When we prescribe a level of singularity other
than that with the Kutta condition, vorticity is also shed without a source
and the field is not zero before the source is switched on, and therefore
this field is not causal in the sense of Jones & Morgan. This, however,
32
is not in contradiction with the reality because the complete problem is
non-linear and the amount of shed vorticity is dependent upon the
amplitude of the source: zero amplitude means zero vorticity, therefore
zero field and thus we still have causality.
~ Fo!'mUZation of the probZem
In the (x,y) -space a compressible inviscid fluid flows subsonically
in the lower half space (y < 0) in the positive x direction with the
dimensionless velocity (1,0) •. In the upper halfspace (y > 0) the fluid
is stagnant, while the two halfspaces are separated by a flat, rigid plate
in y = 0 ' x < 0 and a vertex sheet in y = 0
' x > 0 Everywhere
the dimensionless density is 1 the sound speed M-1 The fluid is
perturbed by plane harmonie waves, coming from upstream in the flow with
wavefronts perpendicular to the plate. A sketch is give.n in figure 2.
stagnant fluid
plate
incident plane waves
c::=> main flow
y-axis
diffracted, and instability/edge-interaction wave ,.,...-
- ·:::.:::: = = -=---: - / unsta bie '~ ~, vortex sheet '~,
.... ~~
primary wave
Figure 2
The dimensionless perturbation quantities are (u,v) for the velocities
and p for the pressure. Later we shall introduce the auxiliary function
v of variable u , but we shall never use the velocities (u,v)
explicitly so this will cause no confusion. The perturbation amplitudes
are considered to be small enough to justify the sound speed being taken
33
as constant and the perturbation of the vortex sheet position being
negligible. The regions containing vorticity are consequently restricted
to lines (surfaces in the complete 3-dimensional space) and we can
introduce a potential ~ ' defined by v~ = (u,v). The Kutta condition
may be satisfied or not. For convenience we write explicitly
~ = ~o + ~ = ~o + ~c + B ~e , where ~o is the incident wave, ~e is
Crighton's U972a) eigensolution (Helmholtz unstable, not satisfying the
Kutta condition, representing shed vorticity), ~o + ~c is the stable
solution (in which no norticity is shed), and ~o + ~c + Bk ~e = ~o + ~k
is for a particular choice B = Bk the solution that satisfies the Kutta
condition. The index k stands for Kutta and the index c recalls the
analogy with the continuous solution of chapter 2. The Strouhalnumber, or
dimensionless frequency, is w the Machnumber, or reciprocal of the
dimensionless sound speed, is M and the Helmholtznumber, or dimension-
less wavenumber, is k = wM In this problem the only lengthscales are
the acoustic wavelength (the quotient of sound velocity and frequency) and
the hydrodynamic wavelength (the quotient of flow velocity and frequency).
Therefore we can take either w = 1 or k = 1 • Yet we leave them
unspecified, because the results of this chapter will be used later in
chapter S, where viscous effects are included, and a third lengthscale
appears which will then be used as the most convenient reference length.
We suppose that the variables do not depend on the dimensionless time
coordinate t in any way other than through the factor exp(iwt), which
will be suppressed throughout.
Linearising, we have the following equations and boundary conditions
(the index x denotes differentiation to x , similarly for y ) :
~o(x,y) 0 in y > 0
exp(- k l+M x) in y < 0
M + k2 ~ 0
) in y > 0 ' (3.3.1)
p - i w ~
34
+
p
k2 lji - 2 i k M lji X - M2 lji XX
- i w $ - iw lji l+M o
0
} in y < 0 '
lji (x,O) y
0 in x < 0 , since the plate is rigid,
"' + 0 when jyj + 00 , since the field radiates outwards,
iwlji(x, +O) - iwlji(x,-0) - ljix(x,-0) in x > 0 ,
(3.3.2)
(3.3.3)
(3.3.4)
(3.3.5)
since there is no pressure difference across the vortex sheet, When
y = h(x) exp(iwt) represents the position of the vortex sheet, we have
i w h(x) lji (x,+O) y
i w h (x) + h (x) x
lji (x,-0) y
} in x > 0 ,
since the vortex sheet is a material surface. The vortex sheet is
(3.3.6)
1
continuously connected to the edge, which will imply that h(x) = O(x 2)
(x f 0) , and when the Kutta condition is applied we will have
h(x) = O(x3/ 2) (x f O) , The net force on the plate is fini te so the
pressure difference across the plate is (at least) integrable, For
convenience we give k an imaginary part (or rather w , as we keep M
real), k = jkj exp(-io) where 0 < 8 < 11 the field being outward
radiating.
3,4 FoPmaZ soZution
To keep comparison simple we shall adopt the nomenclature of Jones &
Morgan, with the extensions introduced by Munt (1977) in his studies on the
circular jet. Introduce the Fourier transf orm of iµ
$ (s ,y) r"' (x,y) exp(-sx) dx ' s complex, (3.4.1)
Define in the complex s-plane the following halfplanes
35
hl . Re s > - sine l+M
Re s < Jkl sine
called the (t) plane
} (3.4.2)
called the e plane
Indices + and will denote regularity in the (t) and e plane
respectively. The strip - (l+M)-l !ki sine < Re s < lkl sine
connnon to the (t) and the e planes ' will be called the strip of overlap.
Following the same lines as Munt (1977) or Morgan (1974), we introduce the
Fourier transform of h
r h(x) exp(- s x) dx
0
(3 .4. 3)
and the Fourier transf orm of the pressure dif f erence across the plate
(multiplied by M), as far as it is induced by field ~
G_(s) r [ik~ (x,+O) - ik~ (x, -0) - M~ x (x,-0)] exp (-sx)dx
We introduce the complex functions
Branchcuts run from ik
-ik(l + M)-l to -ik 00
defined by
+ \;(o)
! Àl (0) 1 k I'
À~(O) \;(o) 1
lkl 2
and -1
ik(l-M)
respectively.
+ ! where Àl (s)=((l+M)s+ik)',
- ! Àl (s)=((l-M)s-ik) 2
,
+ where \2
(s) !
(s+ik) 2
! (s-ik) 2
to ik 00 , and from -ik
The appropriate branches are
exp ( i Î - i l e) ' and
exp (-i ~ - i l e) 4
(3.4.4)
(3.4.5)
and
Transform the equations (3.3.1) and (3.3.2), and solve the resulting
equations for s in the strip of overlap, applying (3.3.4) and using the
36
are negative in the strip. Transform the fact that Im Àl and Im À2
boundary conditions (3.3.3),
integration constants of the
(3.3.5) and (3.3.6), and determine the
recently found transformed solution, to get
l k F+(s)
(- i À2
(s)y) when y > 0 M À2(s) exp
~(s,y)
ik+Ms F+(s) (i Àl (s)y) when y < 0 ---rM • Àl (s) exp
} (3.4.6)
together with the functional equation
G_(s)
where
+~ 1 l+M • --:-n
s l+M
X (s) 1 k \ (s)
k3 - i _M_\_(_s..;.;)-À2_(_s_) x(s) F+(s) (3.4.7)
(3.4.8)
is the crucial function of the analysis. X(s) = 0 is the dispersion
relation for the possible waves in the vortex sheet. X has (for M < 1)
' two zeros: s0
and s1 , corresponding to the' increasing and decreasing
Helmholtz instability waves. For convenience we write explicitly
X(s) 1 (s - s 0
) (s - s1
) µ (s) (3.4.9) k2
and
s - ik u - ik! (ç; +in) 0 0
) sl - ik u1 - ik ! (ç; - in) (3.4.10)
1
where ç; = (1 + (1+M2 )2) M-1 ' n = (2/;;M-l-l)j
The derivation of the expr~ssions for
Morgan (1972).
s and 0
can be found in Jones &
We will now solve equation (3.4.7) by means of the Wiener-Hopf
technique. A good introduct_ l with detailed examples is given by Noble
37
(1958). The basic ideas consist of the application of a few well-known
theorems from analytic function theory in a context where they supply great
power, Ignoring all the possible generalisations, it goes essentially as
follows: consider the complex s-plane, split up in the <±> and the e halfplanes with the strip of overlap in common, Let the unknown function,
to be found, be the pair {4+(s), 4_(s)} where 4+ and ~ are
regular in their respective halfplanes, Suppose {$+' 4_} is related on
the strip by the equation W(4+' ~ ,s)
to write:
0 , and suppose that we are able
are regular
in their halfplanes, Then the identity theorem tells US that W+ and W
define one and the same integral function E(s), If we have, moreover,
extra information about the behaviour at infinity of 4+ and 4
halfplanes, leading to the estimates:
in their
E(s) in the 8 plane)
E(s) o<I si q) <s _,_ 00 in the <±> plane) then from Liouville's
theorem it follows that E is a polynomial of degree less than or equal to
the integral part of the minimum of p and q
negative degree are taken to be identical zero,)
("Polynomials" of
The coef f icients of E
are to be found from boundary or similar conditions, or in order to
preserve analyticity requirements.
The first thing to do now, to prepare the splitting of (3.4.7), is to
write ;J(s) D+(s) / (l_(s) , where µ+ and il are not only regular,
but also non-zero in their respective halfplanes. At least in principle,
this is always possible; for example, we have the representations (an
entire non-zero function can always be inserted as a factor) :
with
--P 27ri
d+ioo
f log\;(s')
s'-s d-ioo
hl sin o < d < Re s l+M
ds' (3,4.11)
Re s < d < lkl sinê for Q_ , and d inside the strip.
38
The capital P means that the principal value may be taken if necessary
(Noble (1958), p. 42 ; see also (3.5.1) ).
When the complex value of k is close to real values (i.e. o small), we
have s 0
E 8 plane and s 1
E ® plane furthermore
- ik (l+M)-l E @ plane, so it follows readily that a splitting of
equation (3.4.7) is then achieved by
G_(s) À~(s) 9s>µ _ (.s) +
s - s 0
(s) À~(s) 9s> A -ik - -ik - -ik
ik 1 [µ _ µ - (l+M ) Àl(l+M) À2(1 +M)j l+M ~
+ ik s-r-!- - s-s
0 -- + s l+M l+M 0
ik(s-s1) IJ+(s) F+(s) A -ik - -ik - -ik
ik µ_ (l+M) À{li:M) À2(l+M) 1 +-- ik ik l+M + À+ (s) À+ (s) l+M s s + l+M M 1 2
0
E (s) ,
where E is an entire function. It is easiest to bring
left because F+(s) must be regular in the (f) plane •
leave s - s 0
in the right hand side of (3.4.12), E (s)
(3.4.12)
s - s to the 0
If we were to
would have to be
chosen in a way to eliminate the pole s = s0
from F+ and the result
would be the same. Standard methods provide the estimates
p+ Cs} = o<lsl !> (s _,. 00 in the G:l plane), il (s) =oei si-!> (s _,. 00 in
the 8 plane) and (from h(x) = O(x!}(:X + o)) F (s) = O(lsl-3/ 2) + 1
(s-+ 00 in the ® plane). It follows that E(s) = O(lsl-) (s-+ 00 ) ,
theref ore E _ 0 • Now it is possible, by reordering the terms of
(3.4.12), to obtain G and F+ explicitly. We see that
G_(s) = O(I si-!) , so indeed the pressure difference across the plate is
integrable. Only F+ is important for the solution ~ , via (3.4.6).
As was anticipated before, in section 3.2, we found here by straight
forward application of Fourier transformation only the Fourier transform
able solution, that is the stable solution ~c , where no vorticity is
shed from the edge and the Kutta condition is not satisfied. This
solution is represented by
F (s) +,c
1
M /2. (2+M) 2
(l+M)2 (u 0
µA -ik
(l+M) (3.4.13)
39
The regular solution ~k is found by adding an appropriate multiple of
Crighton's (1972a) eigensolution Bk ~e (Crighton & Leppington (1974)).
Another, equivalent, way may be analytic continuation in k , starting
from the solution with imaginary k (o = jn) , as was applied by Crighton
& Leppington (1974), and Morgan (1974). (If jn, both s and 0
not elements of the plane , so correspond both to damped waves and
therefore are then included in the solution.) This way involves only a
little calculation and provides also Crighton's eigensolution easily, when
we repeat the arguments leading to (3.4.12) via (3.4.7) with ~o - 0
(Crighton & Leppington). For k imaginary as yet, the result in terms of
F+ is for the regular solution
;; eik) l+M (3.4.14)
ikM/2 (2+M) j
(l+M)2 ik µ + (s) (s-s
0) (s-s 1 ) (s + 1 +M)
and for Crighton's eigensolution
F (s) +,e
+ + Àl (s) À
2(s)
- M ~~~~~~~~-); (s)(s-s )(s-s
1)
+ 0
(3.4.15)
The final solution in the (x,y) domain is found by inverse Fourier
transform of (3.4.6), where the contour of integration runs through the
strip of overlap, and then letting o +
pole s = s0
crosses the contour when
accounted for. It is not difficult to
o. No te that in ~k and ~ 1
0 ~ 4n' and this has to be
verify that the sign changes,
the e
necessary when s0
passes branchcuts, all cancel when both branchcuts are
passed. We thus have
40
l/J(x,y) lirn ~ li+O 211:1.
i 00-o
= 2!i f -ico+o
ioo
f $(s,y) exp (sx)ds
-io:i
~(s,y) exp (sx)ds + ,- [<s-s )~(s,y)exp(sx)] -] o s=s - 0
(3.4.16)
k,e
The strip of overlap is collapsed into the irnaginary axis and now the
contour of integration runs along the irnaginary axis: at the right side of
the branchcuts and pole when Irn s < - k(l+M)-l , and at the left side of
the branchcuts when Irn s > k
At this stage of the analysis it becornes convenient to introduce the
followingtransforrnation and substitutions:
s - iku,
µ(s) = µ(u),
µ+(u)
kw(u) 1
k((l-uM) 2 - u2) 2 ,
lkexp(i!11) (1-(l+M)u}!
Ä~(s) = Ik exp(-ij11) w_(u) = lkexp(-il11) (1+(1-M)u) ! ,
~ Ä2
(s) = k v(u) = k(l-u2)- ,
+ Ä
2(s) Ik exp(ij11)v+(u)
l = lkexp(ij11)(1-u) 2
Ä;(s) Ik exp(-i!11)v_(u) !
= lkexp(-i!11)(l+u) 2
the ® plane in the u-plane is Irn u ~ 0 . the e plane in the u-plane is Irn u :; 0
(3.4.17)
The square roots v+ and w+ are, chosen this way, principal branches.
Note that x(-iku) = A(u) in the terrninology of Jones & Morgan.
Now we have explicitly
for y > 0 :
1
\(x,y):c- /2(2+M)2
(1 +M)2
+
(l+M)2(u __ l_) o l+M
co+i.o
~e (x,y) = _l_ f 't' 21Ti
- 00-i.o
w+(u) exp(-ikv(u)y-ikux)
v_(u) µ+(u) (u-u0
) (u-u1
)
w+(u ) exp(-ikv(u )y-iku x) - 0 0 0
v (u ) µ (u ) (u - u ) - 0 + 0 0 1
and for y < 0
41
(3.4.18)
(3.4.19)
du
(3.4.20)
! /2 (2+M) 2
00+i.o
µ_(l~M) 2;i f (1-uM)v+(u)exp(ikw(u)y-ikux)
du
(l+M) 2
fi(2+M)l
(l+M) 2
- 00-i.o
1 (1-u M) v (u ) exp(ikw(u )y-iku x) () o +o o o
µ_ l+M w (u) µ (u )(u -u )(u --1-)
- o + o o 1 o l+M
(3.4.21)
42
/2 (2+M)! (1-uM)V+(u)exp(ikw(u)y-ikux) 1 du ,
w_ (u) µ + (u) (u-u1) (u - l+M) -oo-i.o
(3.4.22)
Joo+i.o(l-uM)v+(u)exp(ikw(u)y-ikux)
w_(u) µ+(u)(u-u0
)(u-u1
) du
-oo-i.o
(l-u0
M) v+(u0
) exp(ikw(u0)y-iku
0x)
+ ~~=-~~~=-~~~~~=-~~~-=-~ (3.4.23) w_(u
0) µ+(u
0)(u
0-u1)
The multiplicative factor Bk is given by
(3.4.24)
The integration contours run along the real axis in the u-plane, on the
negative interval just below the branchcuts, and on the positive interval
just above.
In the problem considered the~sound waves propagate in the same
direction as the flow, and inside the flow, Therefore the problem may be
not general enough to justify the general conclusions drawn above with
respect to causality and the Kutta condition. However, extensions to
include oblique wavefronts, coming from the side of the flow or the
stagnant region, are easily made, They show that for all angles of
incidence of a wave coming from the still region (essentially the Crighton
& Leppington (1974), and Morgan (1974) problem) the above calculations are
the same, besides some trivial alterations. An:lin the case of a wave
coming from the flow the calculations are the same as long as the point of
reflection on the vortex sheet moves supersonically for an observer in the
stagnant region, That means that the angle of incidence is not too close
to the negative flow directiqn (see equation (3.4.25) below). Otherwise
43
we would have so-called "total reflection" (Mi les (195 7), Ribner (195 7)),
in which case we have to modify the mathematics substantially. Although
we have not proved it, the general conclusions then still seem to apply.
We illustrate the previous arguments with some details. Consider an
incident wave, coming from the flow, of the kind used in the airfoil
diffraction problem (section 2.4)
where
given by
exp(-ik
S sin G s
cos G -M s
cosG -M ___ s_x)
and the angle of incidence
Then equation (3.4.7) becomes
G (s) + ik 1-Mcose
s case - M
s + ik---s __
e. 1
0<8i<TI, is
(3.4 .25)
cos G - M in the strip of overlap sinó s
< Re s < 1k1 sin 6
l 1 Total reflection occurs when - M2 (2+M) 2 < tg 8. < 0 , in which case the
1
strip is empty. Apparently it is not enough merely to subtract off the
primary wave. The difficulty is presumably cured by removing the waves
reflected and transmitted by the vortex sheet, as well as the primary wavP..
This difficulty does not seem to be related to the one encountered by Jones
& Morgan (1972,p.485). They found that when the source receded to infinity
via a ray within a certain wedge, no limit to the solution was found, and
in particular, the plane wave case was not recovered. We, however, did
find a solution for plane waves coming from these angles, and conclude that
the angular and radial limits involved are not interchangeable.
44
~ Approximation of the split [unations
A main object we pursue in both this chapter and the next one (4), is
an explicit expression for the formal solutions, which are, though cat1plet~
not easily interpretable. Obviously this can only be achieved asymptotic
ally. One of the parameters of the problem will be confined to the
neighbourhood of one of those critical points which allow in the limit an
asymptotic expansion of the formal solution in elementary functions.
Several choices are possible. The choice in this chapter (3) will be of
Machnumber close to zero (0 < M << 1), and in the next chapter (4), of
Strouhalnumber close to zero (0 < w << 1). In both,problems the central
snag is the approximate splitting of the function µ(u) = µ+(u)/µ_(u) •
In both cases we shall tackle this problem in essentially the same way.
Therefore we shall explain the basic idea in general in this section, to
have it ready for application later.
It was already noted in equation (3.4.11) that a possible expression
for µ± (unique modulo an integral non-zero function as a factor) is
given by
oo+i.o
log µ±(u) 1
J log µ (u') du' 21Ti p
-oo-i.o u'-u
N+i.o 1 lim J
log µ (u') du' (3.5.1) 21Ti N->oo u'-u
-N-i.o
where u is element of the respective halfplane. It is important to
observe that the contour of integration is not necessarily fixed. From
Cauchy's theorem for integrals of analytic functions, it follows that we
may deform every finite part of the contour, provided we remain inside the
region of analyticity of logµ(u) , and incorporate the contribution from
the pole u' = u whenever crossed. Now if we can approximate µ(u)
in some part D of the complex plane by (say) µ(u) with a relative
accuracy of O(E) (E small) , i.e. µ(u) = µ(u)(l + O(E)) and D is
large enough and sufficiently well-shaped to enable us to deform the
contour of (3.5.1) into one which is completely inside D then we can
construct approximations for µ1
,
that their relative accuracy is also
valid everywhere, and we can show
O(E) Note that it is aften not
necessary to have the approximations valid everywhere, because we only
need them on the integration contour of the complete solution ~ , and,
moreover, we can perhaps apply for ~ the same contour deformation into
region D as used for µ1
45
For definiteness we assume that µ(u) = µ(u)(l+O(E)) uniformly in u
in the whole complex plane, except near one point, the real number u = a ,
lying on the positive branchcuts. When ju-al ~ O(b) , where b << a
(otherwise the region near the origin, between the positive and negative
branchpoints, is included and this makes things needlessly complicated),
we assume 1 µ (u) - iJ (u) 1 /µ (u) to be not smal 1 (enough). Obviously we
take iJ(u) to be analytic and non-zero in all the relevant regions (see
figure 3).
~I U€[
'~" indented part
)ij _ç;
"' E
real axis contour of int rat1on
\ a / branch cuts ' /
reglörl' of non uniformity
Figur>e 3
To start with, we consider u outside the a + O(b) region, hut
inside the €J plane • Deformation of the integration contour into the
one sketched in figure 3 (everywhere the old contour, except for a semi-
circle in the (:!;> plane around a wi th a radius much larger than b ) ,
does not really change equation (3.5.1) :
46
1 211i p
(+i.o -co-i.o
log µ(u')
u'-u du' where the c in the
integral sign denotes the deformation. However, naw we can apply the
approximation log µ = log µ + O(E) , and get, when we bring the contour
back into its original position with the assumed analyticity of µ(u) ,
log µ+(u) 1 211i p
-oo-i•o
1 211i p
-oo-i.o
log iJ (u')
u'-u
log iJ (u')
u'-u
du' + O(E)
du' + O(E)
Therefore, when we write µ also in split form: iJ
µ+(u)(l + O(E)) (uniformly in u in the (tl plane and outside
the a + O(b) region). (3.5.2)
In the case when u is an element of the a + O(b) region, we have to
account for the contribution from the pole u' = u • This pole iscrossed
bath in the deformation of the contour into the position of figure 3:
oo+i.o
log µ+(u) 2!i p f -oo-i.o
log Î.Ï(u') du' + log µ(u) + O(E) , u' - u
and in its deformation back into the original position:
log µ+(u) 1 211i p
and theref ore
oo+i.o
f -oo-i.o
log jJ(u')
u' -u du' + log µ(u) + O(E) ,
il (u)
il (u) µ (u) (1 + O(E)) + i!(u)
(uniformly in u in the (tl plane).
(3.5.3)
47
Note that outside the a + O(b) region (but inside the © plane)
equation (3.5.3) is also valid, being equivalent to (3.5.2), because
- µ ( 0( )) - îJ(l+O(E)) (1 0( )) î1 (1 + O(E)) • \1 +:- 1 + E = \1 + ·- + E = +
µ µ
Furthermore, the restrictions we made with respect to u being an element
of the €) plane are superfluous in the result, as we will see, and only
necessary in the proef and derivation, where usewas made of (3.5.1).
Following the same arguments as in the above, we find for µ_(u) ,
with u in the e plane : µ_(u) = îi (u)(l + O(E)) uniformly for u
in the 8 plane. Via µ+(u) = µ(u)µ_(u) = µ(u)µ_(u)(l + O(E))
µ+(u)µ(u)/îi(u)(l + O(E)) we see that (3.5.3) is realy valid everywhere.
In an analogous way the above results can be found toa for µ , and for a
real number a' (instead of a) lying on the negative branchcuts.
up everything, we have
Summing
for a on the positive real axis
µ+(u) µ+(u)µ(u)/µ(u)(l + 0( E))
(uniformly in u ) µ_(u) µ - (u) (1 + O(E))
(3.5.4)
and for a' on the negative real axis
µ+ (u) îi+(u)(l + O(E))
(uniformly in u ) (3.5.5) µ (u) µ (u)îi (u) /µ (u) (1 + O(E))
When more points a and a' are involved, the extensions are analogous
and obvious, and will not be given here. Equations (3.5.4) and (3.5.5)
may be useful, but in the cases we consider, p is still toa complicated
to permit (3.5.4,5) to be generally applicable. We shall use the simpler
form (3.5.2) in the more general evaluations of ~ (with the contours
indented as in figure 3), and (3.5.4,5) when only a fe~ pole contributions
are relevant.
48
3. 6 Approximation [or small Maah nwnber
Outside the regions u = ± 1 + O(M) it is straightforward to
approximate
w(u) v(u) (1 + O(M))
µ (u) M2 v(u) (1 + O(M))
and, applying section 3.5
v+(u) (1 + O(M))
µ (u) (M2 v_(u)rl (1 + O(M)).
From the exact expressions (3.4.10) it is easily deduced that
u 0
l+i M (1 + 0(M2)) ,
1-i (1 + O(M2)) • M
)
(3.6.1)
(3.6.2)
(3.6.3)
(3.6.4)
(3.6.5)
Note that these only suf f ice in the factors -:-1 (u - u0
, 1) of the
integrands if
hood of u = 1
/ u - u 1 / » M • As u = (1 + M) -l is in the neighbour-o,
which lies on the positive branchcuts, we may apply
(3.5.4); therefore use (3.6.4), and we have
[M2 v_( 1!M>r1 (l+O(M)) = ~ /2 M-2 (l+O(M)) (3.6.6)
The factors of the integrals (3.4.18,21) and (3.4.19,22), i.e. $k and
$c , are then
/2. (1 + 0 (M) ) M2
~ exp (-i}c )(l • O(M)). )
(3.6.7)
As was anticipated before in section 3.5, we deform the contour of
integration for the integral expressions of ~k , ~c and ~e from its
position along the real axis into one on which the ± 1 + O(M) and the
(1 ! i)M-l + O(M) regions are avoided (the regions of non-uniformity).
Another desire, however, on technical grounds, is the deformation of
the contour into the so-called contour of stationary phase.
(outside the flow) this contour is given by the hyperbola
u cos(8+iT) ( T real, - oo < T < oo) ,
and for y < 0 (in the flow) it is given by
u - .!!._ + l cos (G+i T) , sz sz
where we have used some nomenclature of chapter 2, namely 1
x = r case = SX = SR cosG , S = (1 - M2 ) 2 (see figure 4)
)
For y > 0
(3.6.8)
On these
stationary phase contours the arguments of the exponentials of the
integrands are imaginary: -ikux - ikv(u)y ikr cash T ,
- ikux + ikw(u)y = iKMX - iKRcoshT (k SK). Of course, when the
49
poles u = u0
, u1
or branchcuts are crossed, we have to account for that.
We shall try to combine the two ambitions - avoiding the trouble regions
and having a stationary phase contour - as far as possible.
When e is not in an O(/M)-neighboerhood of 0 , TI or - 7T, and
not in an O(M2 )-neighbourhood of l n or - ln the stationary phase
contours are completely outside the regions of non-uniformity, and we can
employ the approximations (3.6.1,3,5,7) for arbitrary kr When 8 is
arbitrary, we have to indent the stationary phase contour in an appropri
ate manner to stay outside the regions of non-uniformity (see figure 4),
50
before we may apply the approximations (3,6,1,3,5,7), But then we may
deform the contour again to have it still in the stationary phase position,
So these restrictions on e are not necessary. There is, however,
another difficulty which naw does introduce a restriction on e For
convenience, we did not use the exact expressions for u 0
and ul hut
rather the approximated ones. This is qui te in agreement with the
accuracy considered, except for the instability wave. In the downstream
wedge n < e < 2':. - 4 4 (approximately), where the instability wave becomes
large, the error O(krM) of its argument yields for kr >> 1 an
exponentially large error in the wave amplitude itself. Hence we must
have in TI TI - 4 < e < 4 the restriction kr ~ 0(1). This is nothing to
worryabout, as far downstream the linear model of a vortex sheet is not
correct in any case. Concluding the above, the approximations of
~k' ~c and ~e we will construct along the lines just sketched, are valid
for kr arbitrary, when lel> !n + O(M2), and kr~O(l) otherwise. (3.6.9)
Note that these restrictions are mainly due to the instability wave, and can
be relaxed as far as the acoustics are concerned. When we ignore the
instability wave , we have kr arbitrary, except when 0 < e ~ O(IM).
Here kr ' 0(1) is necessary, because otherwise the bow wave becomes
important, as amore detailed analysis shows. If required, of course, the
instability wave can be included everywhere by using the exact expression
of u0
(3.4.10) in the argument of the relevant exponential function.
real axis 1
-1-M --------------~,--
Figure 4
e Uo
indented contour of stationary p'iase
After the application of the deformations and approximations, and
introduction of the notations
x
y
k
u 0
r cos8 ex
r sin8
8K ,
cos(e +ii: ) 0 0
8Rcos8
RsinG ,
- M 8-2 + 8-2 cos(G +i'i' ) 0 0
where eo > 0 ' 80 < 0 ' T < 0 0
and 'i' 0
> 0 ,
u 0 * (the complex conjugate of u0 ) ,
we have, for example, in ! '11 < e < '11
2i
sin! (8 +iT) exp (-ikrcoshi: )di:
(3,6,10)
f (cos(B+iT) - cos(9 +iT )) (cos(S+iT) - cos(S -iT )) (cos(S+ii:) -1) 0 0 0 0
The other formulae are analogous and not worthwhile to write down
explicitly. Instead, the partial fractions (the u is used for
convenience, but really the transformed form is meant)
(3.6.11)
51
52
1
1
1- u M
1-u 1-u ! M2 {! i M (--1 ___ o) + _l_}
u-u0
u-u1 u-1
! (1 + i) M (-1- - - 1-) u-u1 u-1
- ! iM (-1- - _l_) u-u
0 u-u
1
_ ! M2 {! 1-i (1 +M) + 1 1 +i (1 +M) 1 } u-u
0 2 u-u
1 ·- --1-
u - l +M (u-u ) (u-u ) (u - - 1-)
o 1 l+M
1- u M
(u-u ) (u --1-) 1 l+M
1- u M
j(l+i)M (i !:!l!_ - --1-) u-u 1
1 u - l+M
_ ! M (-1- + _l_) u-u
0 u-u
1
(3,6J2)
applied in (3,6.11) and the relevant similar forms yield standard integral~
as discussed by Noble (1958), pages 33,45. Following the usual steps, we
find after tedious and toilsome calculations, the desired expressions, in
which we introduce a version of Fresnel's integral (see section 6.1)
F(z) exp ( iz2) f exp ( - i t 2) dt ,
z
and its arguments
where
(kr)! [v (u ) cos!8 + v (u ) sin!8] + 0 - 0
* y denotes the complex conjugate of
) (1.6 .13)
y •
53
When y > 0
1 2 ! 1 ( ) ( . . ) 4 ïf exp i L; 11 - ikr •
* + F(-y_)}
(3.6.14)
! ; (~) exp(ii-11 - ikr).
(3.6.15)
q,e(x,y) ( . 1 . ) exp - i 4 11 - ikr ,
When y < 0
1
i(~)! exp (i Î 11 + iKMX - iKR),
+ 2 /2 F ((2KR) j sinjG) J , (3.6.17)
( . 1 . . ) exp - i 4 11 + iKMX - iKR ,
- i /2. F ((2KR)! sinje)J , (3.6.18)
54
1 M 1 - - - exp(i-4
11 + iKMX - iKR), 2 !IT
Use is made in these derivations of
! s.4 !IT. exp(i!11-ikr) F(s(2kr)2 cosjll),
f exp (-ikr cosh-r) d-r cosj(i-r +il)
s = sgn[Re(cosjil)].
(3.6.19)
Care should be taken in the use of these formulae and in their derivation
when small numbers are expressed as a difference of large numbers.
A global interpretation is easily given. The terms with
F((2kr)!sinj9) represent the diffracted field when no flow is present
(the Sommerfeld .diffraction), and the other terms represent the instability
wave (with the aid of (6.1.4)) - which, on its own, does not produce sound
- , and its edge-scattered sound field. The incident wave is its own
reflection, and the bow wave is not present in our approximation.
Asymptotic values in the still region for kr ~~are found, using (6. 1.3)
and ignoring the in the far field irrelevant instability and bow waves,
1
1 2 ; -1 exp(-i l11 - ikr)(si!le - 2 M sinje) \+<Po 4 (-) (kr) 2
'
) 11 4
(3. 6. 20)
1 2 ! 1 exp(-il11 - ikr)(~ tj>c + <Po 4 (-) (krf' - (l+i)Msinje),
11 4 s1n 2
where the (sinjef1 is from the F((2kr)! sinj9) -term . The other
terms are O(M) and therefore of the same order of magnitude as the error
term. However, we shall not neglect these terms because they reveal
precisely the influence of the vortex shedding (Kutta condition). It
·follows that not too close to the vortex sheet, where sin!e is small,
the application of the Kutta condition yields a reduction of the outer
sound field by terms of order M ,
55
3.7 Comparison with experiments
For several reasons there does not seem to be very much experimental
data available on the subject discussed in this chapter. The jetflow
should be two-dimensional, relatively wide, low-subsonic, clean with a
thin boundary layer, and the source should be external. These conditions
are not very aften met in technical applications. Nevertheless, the films
of Poldervaart et al (1974) and the experiments of Heavens (1979) may be of
some relevance.
In Poldervaart's films an externally produced plane sound pulse hits
a two-dimensional, very clean jet (0 < M < 1.5) from an arbitrary
direction. Not much diffraction, reflection or transmission is visible.
However, most striking is the observation that an instability wave
(concentrated in a point first and then gradually growing into vertices) is
generated only when the pulse passes the edge, regardless of the angle of
incidence. This confirms our statement that only the Kutta condition
(governing the vortex shedding) rules the instability wave. Of course
one can imagine that a cylindrical pulse, hitting the mixing layer, creates
another instability wave toa. However, this is obviously only a transient
phenomenon, because a harmonie instability wave of this kind cannot match
the edge conditions.
Heavens' (1979) experiments consider a cylindrical jet, triggered by
plane sound pulses, generated inside the flow. An important feature
observed by him is that when the pulse amplitude is high enough to expect
a deviation from a fully applied Kutta condition, the diffracted field
outside indeed shows an increase qualitatively like that predicted in
(3.6.20). Quantitative measurements are needed to find out if our
formulae (3.6.20) are really the relevant ones.
3.8 Conclusions
The interaction of a semi-infinite plane vortex sheet with harmonie
acoustic waves has been discussed. The vorticity shed at the edge has
been shown to be relevant, rather than causality. In fact, when the level
56
of singularity at the edge is prescribed, only the Kutta condition solution
is causal. The exact solution has been approximated for small Mach
number. The method employed is worked out for future reference. The
central idea is to deform the contour of integration and then utilise an
approximate integrand, valid everywhere on the contour. The effect of
the application of the Kutta condition seems to be a decrease of the
diffracted outer field; this is indeed observed in practice.
chapter 4
THE INTERACTION BETWEEN A SUBSONIC UNIFORM JET FLOW,
ISSUING FROM A SEMI-INFINITE CIRCULAR PIPE, AND A
HARMONIC PLANE WAVE WITH SMALL STROUHAL NUMBER
4.1 Introduction
Aerodynamic noise is usually an unwanted by-product of flow in
techniccl applications. Of all the kinds, aircraft noise is probably
most annoying, as it involves by far the most acoustic energy and is
essentially produced in open air. Therefore (to prevent the growth of a
public antipathy against airplanes) the greatest need was for research on
jet noise in particular. As a consequence, most of the general
57
theoretical results in aeroacoustics are only tested for jet flow. It is
not always possible, however, to distinguish the considered theoretical
aspect in a complex experimental situation. We can try to "clean" the
experiments then, but even so it can still be difficult to decide whether
the mechanism studied plays its imputed r6le or not. Then we have to
attempt an extension of the theoretical model in the direction of the
experiments, thus inevitably involving a jet. An example with typically
a history like this, is the problem of the interaction of sound with a
mixing layer started by Miles (1957) and Ribner (1957) and resulting in
Munt's (1977) work. This chapter will provide a further step on the path
shown by Munt.
A large part of the acoustic energy emitted from a jet comes from the
turbulence. The moving, collapsing and interacting turbulent eddies act
macroscopically as an acoustic quadrupole distribution (Lighthill (1952),
(1954)), and indeed in an experimental situation at model scale the jet
noise follows qui te perfectly Lighthill's predictions, at least in respect
of the famous US law for total radiated power. However, in the operation
of a jet engine, the noise appears, for certain frequency bands, to exceed
the theoretical values. These deviations have probably several sources,
but altogether they are called "excess noise" (Crighton (1972b)).
58
Crighton (1972b) conjectured that, a~ least partly, the hydrodynamic
instabilities of the mixing layer are responsible for this, and he
calculated expressions for the Helmholtz-type instabilities on a semi
infinite cylindrical vortex sheet, without the Kutta condition applied.
However, qualitatively the cylindrical case is not so different from the
plane case, and therefore, as we know now (chapter 3), these instabilities
do only develop when driven by an external source , and Crighton's theory
had to be extended to include some source before comparison with experi-
ments was possible. But at that time, 1972, the understanding of the
interaction of sound with an infinite plane vortex sheet was just budding,
and with a semi-infinite plane vortex sheet still on the stocks (see
chapter 3), so one had to wait a while.But after the appearance of these and
subsequent investigations - of which we mention the work of Morgan (1975),
who calculated the field of a point source inside an infinite cylindrical
jet flow - Munt (1977) collected all the basic ideas and solved the (up to
now) most general model of sound - mixing layer interaction.
Munt considered a semi-infinite circular pipe from which issues a subsonic
inviscid jet in a subsonic co-flowing (slower than the jet) inviscid
medium. The sound speeds, mean densities an.d main stream velocities in -
and outside the jet are arbitrary. The jet is perturbed by sound waves
coming from inside the pipe. The Kutta condition is applied, and the
instabilities are allowed to grow exponentially unlimited as they convect
downstream. Also causality was required, but as we saw in section 3.2,
this aspect can be ignored. Munt calculated very carefully the
behaviour of the zeros of the dispersion relation, identif ied the right
poles, and calculated numerically the far field for several choices of the
parameters. Now, for the first time, a comparison with experiments could
be made to estimate the importance of the interaction of the internal
sound (in practice generated by the combustion and the turbo machinery)
with a mixing layer (i.e. the relevance of the sound generated by the shed
vorticity). The results were, at least for the cold jet, remarkably
good. The Kutta condition played a significant röle. If no instabili
ties and their associated sound f ields were included the outerf ield was
definitely not in accord with the measured values.
In an attempt to participate in this success, we shall evaluate in
this chapter Munt's formal solution analytically in an as~mptotic expansion
for small Strouhal numbers w = ~fD/U , where f is the incident wave
frequency, D the pipe diameter, and U the jet velocity. As a start,
we take the particular case of no external flow, equal mean densities and
sound speeds inside and outside the flow, and the zeroth (axisymmetric)
mode of the incident sound wave. As a next step, it is possible to
consider amore general case by relaxing these assumptions, but this has
not been done here. It may follow in the future, because it would ease
comparison with experiments.
59
Employing the technique described in section 3.5, we shall calculate
the pressure and axial velocity inside the jet as an infinite series
(reducible on the jet axis to more or less elementary functions), and also
the pressure outer field far from the jet. Differentiation of the
approximated field inside the jet appears to be risky, and therefore we
have to consider the variables (pressure, velocities) separately there.
We shall only consider the Kutta condition solution, because Bechert and
Pfizenmaier (1975) showed experimentally that (in the range of amplitudes
they considered) the Kutta condition is always satisfied for small
Helmholtz numbers k = WM. Of course, as we saw in section 1.3, for
higher amplitudes a singular solution may be more appropriate. We note
We note that Howe (1978) recently considered aspects of the present
problem in a particular context. He studied the interaction of sound with
different jet flows from various types of nozzles to investigate the
attenuation of the radiated sound power observed in practice for low
Strouhal number. Howe does not give many details, hut his conclusions
seem to be consistent with ours.
We shall basically rely on Munt's results and therefore only briefly
derive his formal solution in integral form. A straightforward way would
be to calculateviaFourier transformation the Fourier·transformable (i.e.
stable, without vortex shedding) solution, and then add one of Crighton's
60
(1972b) eigensolutions to apply the Kutta condition. However, the
equivalent method employed by Munt of'making both relevant (Helmholtz)
instabilities damped (giving k imaginary values) and then obtaining the
real k solution by analytic continuation requires only little mathematics,
is more elegant and will therefore, as in chapter 3, be used here.
Beside this remark, we note that the problems concerning causality and
Kutta condition are the same as in chapter 3, and are fully discussed in
the sections 3.1 and 3.2. They are therefore implicitly accounted for
without much discourse.
4.2 Fol'171UZation of the probZem
We consider the problem of a semi-infinite cylindrical rigid pipe,
exhausting a jet flow with a top-hat velocity profile, interacting with
plane harmonie sound waves, also coming from inside the pipe. The fluid
outside is at rest relative to the pipe, and thermodynamic properties such
as mean density and sound speed are equal in- and outside the jet. The
fluid is inviscid and conducts no heat, and the governing equations will
be linearised with respect to the acoustic and hydrodynamic disturbances
(see figure 5 ).
diffracted, and instability/edge-interaction wave
(
---------z-axis
---- - -----we~iY lÏiStäbTê-vortex sheet
stagnant fluid
Figure 5
The explicit analytic result derived finally is an aymptotic series
expansion, for small Strouhal number, of the axial velocity and the
61
pressure inside the flow, and of the pressure far from the jet.
use the terminology of Munt with a few obvious simplifications.
We shall
Make the problem dimensionless in the following way:
velocities are scaled on the undisturbed jet velocity U, to get
(1 + u,v,w) and (u,v,w) in- and outside the jet respectively;
density is scaled on the mean density p0
, to get 1 + p
lengths are scaled on the radius of the pape jD
time on !D/U and
pressure on P u2 0
2 -1 to get p 0
(p 0
U ) + p , where p 0
is the mean pressure,
The dimensionless sound speed will be denoted by l/M ,
is the Machnumber and c0
the (mean) sound speed; M
where M = U/c 0
is supposed less
than one. As the regions of non-zero vorticity are confined to surfaces
we may introduce ~ , the potential of the velocity disturbances, this
existing everywhere except at the pipe - and jet surface. Here the
required values must be understood as the appropriate limits. Introduce
cylindrical po lar coordinates (r,8 ,z) in such a way that r = 1 , z ~ O
describes the cylinder surface and r = 1, z ~ 0 the undisturbed vortex
sheet separating the jet flow from the ambient stagnant fluid.
For the primary wave ~o , coming from inside the pipe, we take
~ (r,e,z,t) 0
~ (z) exp ( iwt) 0
exp (- i ~ z + i w t) ( r < 1) 2rr l+M
(4.2.1) 0 (r > 1)
where k = w M. If an incident pressure of amplitude unity is preferred,
the resulting solution should be multiplied by 2 rri (l+M)/w . Note that
the dimensionless frequency w is equal to the Strouhal number rr f D/U ,
where f is the real frequency; the dimensionless wavenumber k = w M is
equal to the Helmholtz number We postulate some natural
62
damping mechanism, not actually included in the model, which prevents other
time and angle 8 dependence to óccur than generated by the primary wave.
So in our case the time dependence appears in the solution as the factor
exp (iwt) which will be suppressed throughout, and moreover the solution
is independent of a • Anticipating the use of Fourier transforms, it is
advantageous to write $ as the sum of the primary wave $0
and a wave
W , accounting for the whole set of effects produced by the interaction of
$0
with the pipe edge and the vortex sheet, such as diffraction,
transmission, reflection, refraction, vortex shedding and instability
generation. If we add, temporarily, enough artificial damping, by means
of a non-zero imaginary part of wavenumber k , to make the instabilities
decreasing, W is square integrable, so its Fourier transform exists (as an
ordinary function). After having found the solution for complex k we
obtain the required solution with real k by analytic continuation of the
solution as a function of k See also section 3. 2 •
Taking into account the above-mentioned considerations and applying
the acoustic approximation P = M2p , the equations, with boundary
conditions, for W(r,z) = $ - $0
and p are (the index z or r means
dif f erentiation with respect to that variable)
Wzz + .!. (r W ) + k2 W - 2 ik MW r rr z 0 r < 1 , (4.2.2)
w zz +
1 (r W ) + k2 w
r rr 0 ' r > 1 , (4.2.3)
p w z r < 1 , (4.2.4)
p r > 1 . (4.2.5)
From the rigidity of the cylinder it follows that
0 z ~ 0 ' (4.2.6)
while from the continuity of pressure across the vortex sheet it follows
that
ikl/J(l+O,z) - ikl/J(l-O,z) - M l/!2(1-0,z) 1. k ( )
l+M q, o 2 '
The vortex sheet is a material surface, so that if
r 1 + h(z) exp(iw t) for z > 0
is the position of the vortex sheet, it follows that
i w h(z)
iw h(z) + h (z) z
z > 0
z > 0 •
63
no. (4.2.7)
(4. 2 .8)
(4.2.9)
(4.2.10)
At the edge of the cylinder we impose the Kutta condition of smoothness
h(z) = O(z312
) (z t 0) . (4. 2 .11)
The field of ljJ is radiating outwards, therefore
k 1 ki exp ( - i 6 ) 0 < 6 < 1T • (4.2.12)
4.3 Formal solution
Following Munt we introduce
the Fourier transf orm of l/J
S(r,s) J ljJ (r,z) exp(- s z)dz , s complex (4.3.1)
in the complex s-plane the halfplanes
the © plane Re s > - M sin l+M
the G plane Re s < !ki sin 6 '
l (4.3.2)
where the common strip is called the strip of overlap; the indices + and
64
- will denote regularity in the respactive halfplanes;
the Fourier transform of h
F+(s) = J h(z) exp(- s z) dz
0
(4.3.3)
the Fourier transform of the pressure difference across the pipesurf ace as
far as it is induced by ijl
G_(s) ( [ikljl(l+O,z) - ikljl(l-0,z) - Mwz(l-0,z)]exp(-sz) dz
and the complex functions
1 À
1(s) = (s2 - (ik+Ms)2) 2
+ Àl (s) ((1 + M) s
À 2 (s) (s2 + k2) ! + À 2 (s)
+ ik)!
À;(s)
+ À2(s) (s +ik)! , À; (s)
where
ÀÎ (s) ((1 - M) s - i k~
where
(s-ik)!.
(4.3.4)
(4.3.5)
Branch cuts run from ik and ik/ (1-M) to ik "' , and from -ik and
-ik/(l+M) to -ik"' respectively.
by
The appropriate branches are def ined
! J kJ 2 exp ( i ! 11 - i ! ê ) , and
À~(O) 1
JkJ 2 exp(-i ! 11 - i ! ö) • (The same as (3.4.5)).
Introduction of (4.3.1) in (4.2.2) and (4.2.3), and application of the
boundary conditions leads to
tl (r, s)
tl(r,s)
(ik+ Ms) F (s) +
ikF (s) + H <2> {À (s)r)
M À2 (s) Hp) (À2
(s)) 0 2
r < 1
) (4.3.6)
r > 1
and the functional equation
( ) + _!__ i_l;_ 1 G s 2Tr l+M ik
s + l+M
k3 F+{s)x(s)
M Àl {s) À2
{s)
65
(4.3.7)
all valid for s in the strip of overlap. The function x, defined by
X (s) - (ik + Ms) 2 À
2{s) J
0(À
1{s))
k3 Jl (Àl (s))
À (s) H(Z)(À2
(s)) 1 0 (4.3.8)
is crucial in the analysis. The dispersion relation of the possible waves
in the system is given by x(s) = 0 . The functions Jn and H~2 ) are
the n-th order Bessel function of the first kind, and the n-th order Hankel
function of the second kind (Watson (1966)).
Equation (4.3.7) will be solved by means of the Wiener-Hopf technique
(see Noble (1958), or section 3.4 right after (3.4.10)). The first thing
to do, to prepare the splitting of (4.3. 7) is to write x = x+I x _
where and x_ are not only regular but also non-zero in their
respective halfplanes. So the zeros of x play an important róle and
must be treated carefully. According to Munt's analysis (partially
relying on a numerical search) there are, from the infinity of poles and
zeros, just two zeros, denoted by s and 0
plane into the other for varying complex k
which move from one half-
Therefore it is convenient
to write (slightly different from Munt, to be uniform with chapter 3)
1 x{s)= --(s- s ) (s- s3) µ (s).
k2 0 (4. 3. 9)
We note that in the case of small w (therefore small k ) the
behaviour of the zeros will be obvious, and we do not really need Munt's
numerical work. At the end of this section we shall summarise the
general properties of the zeros, but of these it is now important to note
that when
66
I.
II.
the
is in the neighbourhood of 0 , s0
is not an element of
e plane ' and s3 is not an element of the ® plane,
is in the neighbourhood of
elements of the ® plane.
w, both s0
and s3
are not
A splitting il = (l+!(l_ with µ+ and µ regular and non-zero in
their halfplanes, can be made now independent of k in general behaviour,
from which the split functions X+ and x follow by inspection with the
aid of (4.3.10). Explicit expressions of µ± are always possible (see
(3.5.1)) but we do not need them here. Using the splittings of +
and +
and noting that - ik/ (l+M) is element Àl = Àl Àl À2=À2À2 ' not an
of the ® plane, we can establish the splitting of (4.3. 7) for the two
cases of (4.3.10). As was anticipated, case I yields the stable
solution, which cannot satisfy the Kutta condition, so we shall ignore
this. Case II gives
G (s) À~(s) À; (s) µ (s) + -
ik [À~ (s) À; (s) ' (s) - -ik - -ik ' -ik ] 2iT l+M. ----rk µ - Àl (l+M) À2(l+M) µ_ (l+M)
s + l+M
1 ik 1 - 2w 1 +M ik
s + l+M (4. 3 .11)
The equality holds in the strip; the left hand side is regular in the
e plane, the right hand side is regular in the ® plane, and so they
define together an entire function E(s) . To determine this function we
use some well-known results (the "Abelian theorems", Noble (1958), page 36)
which relate the behaviour of a function near the origin to the behaviour
of its Fourier transform near infinity. From the Kutta condition (4.2.11)
it follows that F+(s) = O(s-512 ) when Is! ~ oo in the ® plane.
Secondly, the force on the pipe surf ace is finite, so the pressure
dif ference is at least integrable, i.e. O(z-l-q) for z t 0 with q < 0,
which gives G (s) = O(sq) when - 1 si ~ 00 in the e plane. Furthermore,
67
' l applyjng standard methods, it can be deduced that µ+(s) = O(s 2
) and
µ (s) = O(s-!) when 1 si ~ in the respective halfplanes. These
estimates result finally in E - 0 . We note that if the Kutta condition
is not applied, we have F+(s) O(s-312
) and then E is a constant.
After the subsequent calculations the factor of E is the relevant
eigensolution of the set found by Crighton (1972b). In relation to this
it is now easily seen that the theorem, stating that there does not exist
without external excitation a Helmholtz instability wave in a vertex sheet
originating from an edge and satisfying the Kutta condition (formulated by
Bechert & Michel (1975) for the incompressible - and by Howe (1976) for
the compressible two-dimensional case), is valid also for the cylindrical 1 ik 1
case: 2n l+M -------riZ
s + l+M
take equation (4.3.11) with the sourceterm
equal to zero, then from the above it fellows readily that application of
the Kutta condition, i.e. E = 0 , results in the trivial solution
F+ = 0 only.
Equation (4.3.11) can be solved immediately, and we have
+ i
(4.3.12)
Equation (4.3.12), corresponding to (5.3) in Munt's paper, supplies, after
being substituted in the inverse Fourier transform of (4.3.6), by analytic
continuation in k for 6 ~ O (where the transition of s across the 0
integration contour must be accounted for) the following solution (see
equations (5.7a) and (5.7b) in Munt's paper)
ij; (r, z) 1
2ni
ico-o
J -ico+o
(ik+ Ms)F + (s) J0
(),1
(s) r) ----,--,,----0----,-,----,-'7-- exp ( s z) d s
M Àl (s) J1
(À1
(s)) +
F+ (s) 1 exp(s0
z) , for - s=s
0
(4.3.13)
r < 1 ,
68
ioo-o
1jl (r, z) 1
2î!i J ik F+(s) H!
2)(1,
2(s)r)
M À2
(s) H(Z) (À (s)) exp (sz)ds +
-ioo+o 1 2
(4.3.14)
for r > 1.
The branchcuts coincide now (for real k) with the imaginary axis, while
the contour of integration runs for negative imaginary arguments on the
right side, and for positive imaginary arguments on the left side of the
cuts. We shall call the integrals of (4.3.13) and (4.3.14) the stable
part of the solution, and the remaining exponentials the instability part.
In the next section we shall calculate the far field of the pressure
p - iw1jl '
and the complete field inside the jet of the pressure -1
i w 1jl- and the axial velocity p - iw (l+M) cj> - 1/iz -1 0
cj>z = - i k(l+M) cj>0
+ 1"z For the outer field it is irrelevant whether
pressure, velocity, or whatever variable is taken. Inside the jet,
however, we have to be very careful with differentiating the asymptotic
expansion it changes the order of the terms and thus the order of
magnitude of the error term. It can easily be deduced from the final
expression that in cj> the diffraction effects are of the order of
magnitude of the error term, while in cj>zz the diffraction effects are
part of the leading order term, together with the primary and the
instability wave. The basic reason for this is that essentially two
length scales are involved: the acoustic and hydrodynamic wavelengths
2îl/k and 2î!M/k = 2î!/w , respectively, on the one hand, and the much
smaller pipe radius ( = 1) on the other. The instability wave and the
reflected and transmitted primary waves are longitudinal and propagating
along the z-axis with only slow variations in z and therefore disappear
w'ith differentiation with respect to z • The diffraction waves, however,
are pointing in all directions and give, via reflections against the
pipesurface, order one variations in the z-direction, which preserve the
order of magnitude of the diffraction waves after differentiation.
This completes the derivation of a formal solution to the problem.
Before proceeding with an explicit approximation, we sum up the general
properties of the zeros of x .
As discussed extensively by Morgan (1975) and extended by Munt with
numerical examples of representative cases, the zeros of x are
s and 0
correspondi.1g to the increasing and decreasing HelmholtZ
instability,
69
and near the hranch points, for k + 0 exponentially close to
s = ±ik
an infinite sequence of ze ros near the line
Im s = kM/ (1-M2 ) in the G plane,
an infinite sequence of ze ros near the line
Im s = kM/(l-M2 ) in the e plane
The characteristic behaviour of the zeros as a function of k is (Munt,
p.621), that s0
and s3
are more or less linear ink and the others are
rather insensitive to changes of k they never move to another halfplane
For the limit case w + 0 (sok+ 0) , which we will discuss, this is also
readily clear from the explicit approximations. Of these zeros, s0
, Ç+
and a1
, a2
, ... , with positive real parts, are all candidates to
represent instability waves. However, ç+ and most of the a1
, a2
, ...
correspond to incoming waves and must therefore be rejected. For M small
enough and k not too small, there is a finite number a1
, ... , aN with
negative imaginary part, and so corresponding to outgoing waves, as Crow &
Champagne (1971) and Lee & Jones (1973) found. However, the calculations
show that only s0
and none of the a1
, ... , aN appear as instability
waves. Of course, we are relying here on some uniqueness assumption for
the solution. The physical mechanism for this is not clear as yet.
possible conjecture might be, that the waves corresponding to a1
,
A
a N
cannot be connected to the pipe edge without vanishing everywhere, and are,
as they do occur in the infinite vortex sheet model, only appearing in a too
far simplified problem. They may occur then, in reality, far from the edg~
hut probably do not because of the thick and turbulent mixing layer. No te
that ai and ç± are rather insensitive to k and only scale upon the
70
nozzle dfameter and so it will hardly help to use a bigger nozzle or
shortwave disturbances.
4.4 Approximations, [or small Strouhalnumber, [or the axial veloaity and
pressure inside the flow, and [or the [ar field.
In this section we shall derive an analytic expression for the
solution, asymptotically valid for w + O (and so k = wM + 0), with a
relative error O(w2 log(l/w)). Although the calculations were originally
set up with the restriction of a small Machnumber, it became clear that
this was not necessary within the accuracy considered. The only limitati.ons
are to have 1-M2 = 0(1), and M :t' wn for some positive n. We
start with the field inside the flow. Some of the results obtained are
then afterwards also used for the far field. At least in the first
instance it eases the calculations very much if, inside the jet, we
restrict the range of the z-coordinate: z :; 0 (l/w). (4.4.1)
This means that the instability wave amplitude has not yet grown signifi!
cantly. The finite Machnumber is often met as the factor (l-M2 )2
,
therefore we introduce S = (1 - M2) ! . For simplicity we introduce the
following transformation and substitutions (see also (3.4.17)
s -iku
x(s) x(-iku)=x(u)
F+(-iku)= F+(u) (4.4.2)
k w (u) k((l-uM)2 - u2)! = kw+(u) w_(u)
k v (u)
We take v, v ±. w and w the same ±
as used in chapter 3, and as used
by Morgan (1975), but not the same as those used by Munt (1977). We
choose v± and w± as principal branches, so v(O) = w(O) = 1 and we
have branchcuts from and l/(l+M) to and from -1 and
-1/(1-M) to on the real axis. The <±> plane in the u-plane covers
now (with cS = 0) Im u 'il 0 '
and the e plane Im u ~ 0 The contour
71
of integration runs from + 00 above the branch cuts to underneath
the branch cuts, crossing the real axis somewhere between -1 and 1/ (l+M).
As might be expected, the behaviour of the integrands of (4.3.13,14)
is not uniform on the contour of integration for w ~ 0 . If we stick to
the relative error of O(w2 log(l/w)), however, it is almost uniform, apart
from a few points which can be avoided, using section 3.5 under the
condition (4.4.1). Otherwise they must be treated separately (section4.5).
The real prnblem in approximating the integrands is the approximation of
So we shall start with this. We have
x(u) J (kw(u))
(1 - uM) 2 v(u) 0 - w(u)
J1
(kw(u))
H( 2)(kv(u)) 0
The expansions for small arguments of the Bessel functions give
(Gradshteyn & Ryzhik (1965))
J (z) 0
Jl (z)
H( 2 ) (z) 0
2 (1 + 0(22)) z
- z log (j i z expy)(l + O(z2 log z)) ,
)
(4.4.3)
(4.4.4)
(where y = 0.5772 ... Euler's constant, and the principal branch of log
is taken), and for large argument, when lul ~ 0(1/k) on the contour
J (k w (u)) I (Bku) 0 sgn(Im u) 0 + O(wM2 ) 0 ( 1), ----
J1
(kw(u)) r1
(Bku)
(4.4.5)
H( 2 )(kv(u)) K (ki ui) 0 0
0(1) ' Hi
2) (k v(u)) - 1
Kl (klul)
where I and K are modified Bessel functions of the first and second n n
kind. Note that 1
w(u)/B v(u) ~ (-u2) 2 - i sgn (Im u) u that on the contour Im u
72
changes its sign together with Re u , that In/Im , K0
/K1+ 1 for
lul + oo on the contour, and that In has no zeros outside the origin, and
Kn has no positive zeros or singularities (Watson (1966)). Using (4.4.4)
we can find approximations to the most important zeros u 0
i s /k 0
ul = i s3/k viz
1 (1 + O(w logi ..!.) = ..!.o + i ! II wlog! ..!.)(1· + O(w log-! 1 )
u M ;». 0 w . M w
1 (1 + O(wlog! -b> = ..!.o - i ! 12 wlogi ..!.) (1 + 0( w log-! ..!.)) • ul = -M M w w
and
(4.4.6)
In this section, we will only use the rougher approximations. However,
these have zero imaginary part. To distinguish them and to realise that
the contour of integration runs between them, and that u0
is an element
of the G> plane and ul of the e plane' it is important to refer to
the imaginary part of the exact u0
and u1 . We shall do so by writing
1 i. 0 u -+ 0 M
1 i. 0 ul M l (4.4. 7)
Using (4.4.4,5,7), and after analyzing the regions near u = +I and u = -1
in particular, it can be shown that an approximation to x is given by
M2 (u 1 M
J (kw(u)) i. O)(u - ..!. + i. 0) v(u) _o ___ _
M J1
(kw(u))
with x x (1 + O(w2 log(l/w)) everywhere, except in the regions
± 1 + o<..l. 0(1) (loosely speaking)
} u exp(- --))
' k2 k2M
1 (1 + O(w log! l)) u "M w
(4.4.8)
(4.4.9)
Outside these regions, we know from section 3.5 that X+ and x , where
~ = x+/x_ , approximate X+ and X_·
decomposition of J0
and J1
Using the infinite product
73
J (z) 0
z2 ll (1 - --)
n=l ·2
) 3o,n
(4.4.10)
1 z2 2
z ll (1 - -.2-) n=l 31 ,n
where jo,n' jl, n is the n-th zero of J o' Jl (Watson (1966)), we can
construct the following expressions for x+ and x
M j2 62 !
v + (u) u + +- (~- 1)
j x+ Cul (u -~- i.O) (u -~+ i.O)
s2 s2 k2 ~ w+(u)
ll j2 62 ! jo,n n=l M i ( l ,n u + +- - 1) s2 s2 k2
(4. 4 .11)
j2 62 1 M i 2
(u) j u + (~- 1)
k w s2 s2 k2 (4.4.12) x (u) -
ll .....2..>E. 2M2 v (u) n=l \,n M i j2 62 !
u + (~ - 1) sz s2 k2
Note that these are of algebraic behaviour at infinity.
If, following the ideas presented in section 3.5, we deform the
integration path for the integral expressions of q, and z
p in to the one
described in figure 6 '
we can use approximations (4.4 .11) and (4.4.12).
The new path consists of the old one with an indentation around u = l/M
forming a semi-circle in the (tl plane with a radius many times -1 l
w M log' (l/w), and indentations around u = +I and u = -1 with radii
largely depending on M. When M is of order 1, the radii may be of the
order of some positive power of w ' whereas for smaller M they may be
exponentially small. In fact, the regions near u = +I and u = -1 are
of a difficult type. We have an infini te, nested sequence of regions of
non-uniformity there. HoT'lever, they can all be avoided, and we shall not
discuss this point any further.
74
uet
.~
\ \, .Ut,/
region èr rm-oofamity
·Figure 6
The pole u = u0
is crossed during the deformation, so the instability
parts of (4.3. 13) and (4.3. 14) are included then in the integral.
Note that we need the restriction (4.4.1), i.e. z ~ 0(1/w).
Approximated as in (4.4.7), u0
yields for z > 0 an error of
O(zw2log!!) in the argument of the instability wave, and hence, when w
z ~ 00 , an exponentially large error in its amplitude. It is a subtle
point that it is necessary only to take z ~ 0(1/w) , rather than
z < O(log! (l/w)). Considering only the influence of u0
, we would expect
the strenger restriction. However, is close to and almost its
complex conjugate, i.e. u0
~ ui giving rise to a sart of interference
between the increasing (u0
) and decreasing (u1
) wave. The resuü is
O(w2 log!(l/w)z}. a relative error of O(w4 log(l/w) z ) , instead of
Illustrative for these points are the following model integrals
( where a corresponds to l/M, and € to jlïwM""llog!(l/w) ):
_ ____,_( u_-_a_),__e_x_.p_,(~-:""i~k-=u=-z ,_) __ du
(u-a-iE)(u-a+iE)(u2+1)
1 f (u-a)2exp(-ikuz) -- du 2ni ~(u-a-iE)(u-a+iE)(u2 +1)
75
for ~z and
for p
where the "c" in the integral sign denotes an indented integration path.
On the other hand, examination of the expression for h ,
h(z) = 2kn f F+(u) exp(-:ikuz)du - ik[ (u-u )F (u)] exp(-iku0
z) , . 0 +
u=u 0
reveals that h contains in the leading order the factor l/(u0-u
1),
which is infinite when we use approximation (4.4.7). Still h is finite
when w ~ O , because the numerator multiplying this factor approaches
zero at the same rate. When we take care, the present approximation is
applicable, but we shall not evaluate this here.
After substitution of (4.3.12) for F+ and -iku for s , we
arrive at the following expression for the axial velocity inside the pipe
and the jet
~ (r, z) z
ik exp(-i_!_z) - 2n(l+M) l+M
+ ikM ( 1 ) 2n(l+M) w_ l+M
00+i.o
2;i f - 00-i.o
(u_l_)u J (krw(u)) v (u) w (u) exp(-ikuz) __ M __ ~o __ ~ __ + ___ + ________ du .
w(u)(u--1-) J
1(kw(u))x+(u) l+M
(4.4.13)
Substitution of (4.4.10), (4.4.11) and (4.4.12) gives <ignoring the terms
irrelevant in the present approximation)
76
where CJ
ik -ik iw exp(l+M z) +
7r(l+M)f32 (1 . k )
- 1 j3 CJ 27r (l+M)
u J 0
(krw(u) )exp ( -ikuz)
(u -.!.) (u --1-)(u +-1-) M l+M 1-M
+ O(k w2 log.!.) , w
l .,]_ - -. l_ n=l Jo,n Jl,n
0.2554 ...
(4.4.14)
(4.4.15)
When z > 0 we can close the contour via a large semi-circle in the lower
halfplane, and when z < 0 in the upper halfplane, and obtain in the same
accuracy
z > 0 ik ( 1-M l - 7r(l+M) exp -iw(z-(l+M) CJ)]
(4.4.16)
z < 0 ik [ - 27r (1 +M) exp
wk z 2 1 - 7r(l+M)f3 ;<r,8) + O(k w log;>, (4.4.17)
where L0
and Ll are given by
77
J (r j o ,rn ) exp (-z 'j )
i;0(r,z') I 0 o,m (4.4.18)
j j j m=l j (1 + .o,m) Tl (1 - . o,m) (l + .o'm)
o,m J l,m n=l Jo,n J l ,n
n#m
Jo(r jl ) exp(z' j ) i:
1 (r, z') I m l,m ( 4. 4. 19) j j j
m=l j l,m (1 + .1,m) Tl (1 + .l,m)(l _ . l ,m)
Jo,m n=l Jo,n J l ,n n#m
Note that l:0
and i:1
are exponentially small for z' ~ 00 and - 00 respectively.
The analogous result for the pressure p is
p(r,z) iw - 2TI(l+M)
ikM - 2TI (1 +M)
00 +i.o
2~i 1
-ik exp(l+M z)
1 2 . (u-M) J
0(krw(u)) v+(u) w+(u) exp(-1kuz)
- 00-i.o w(u) (u --1-) Jl(kw(u)) x+(u) l+M
Substitution of (4.4.10, 11, 12) yields
du .
(4. 4. 20)
78
p(r,z) iw
- 211(l+M) -ik
exp<l+M z) iw
11 (1 +M) B2 (1-i!.c )·
B J
J0
(krw(u)) exp(-ikuz)
(u +-1-) (u - _l_) 1-M l+M
+ O(w 3 log .!.) , w
and so for
z > 0 p(r,z) wk l: ( z) + 0 (w 3 log .!.) 11(l+M)B o r,S w
and for
z < 0 p(r,z)
+ ~ i: 1 (r,~) + O(w 3 log -w1) 11 (1 +M) B "'
(4.4.21)
(4.4.22)
(4.4.23)
As is usual in duet acoustics, the incident primary wave is cancelled
outside the pipe (z > O); a conclusion is that the presence of the vortex
sheet is not enough to keep more inside the jet than a part of the
diffracted wave.
Outside the pipe the presence of the flow îs obviously marked by the
instability wave. Inside, however, we also observe striking differences
from the no-flow case although these are not as pronounced as the instabi-
lity wave. If we def ine the reflexion coef f icient R as the ratio of the
coeff icients of incident and reflected primary wave in pressure form, we
have from (4.4.23)
79
R c
- exp ( - 2 i k s3) instead of - exp (- 2 ik ,q , (4.4.24)
and an end correction (phase shift due to the reflection)
instead of i 0.6133 ... (Noble (1958), p.115,138). (4.4.25)
For a proper understanding of these results, it is important to note, that
(after performing analogous calculations as above) the pressure wave far
upstream inside the pipe, for the case in which no vorticity is shed from
the edge, is given by
So now the reflection coefficient has IRI = (l+M)/(1-M) , instead of
unity, although, on the other hand, the end correction is still S CJ . As
a conclusion we have that the Kutta condition does not affect the end
correction (at least, when compared with the no-shed-vorticity-case), but
does change the reflection coefficient. The difference is probably best
expressed by considering the power flux. According to Blokhintsev (see
Candel (1975)), the net power transmitted along the tube toward the exit
is (modulo some dimensional constants) given by P "-' (l+M)2 p~ - (l-M) 2 p:_ ,
where and P_ are the amplitudes of the incident and reflected
primary wave. We see that when no vorticity is shed, no power (in the
limit k + 0) is transmitted, in contrast to the Kutta condition case,
where P "-' 4M /(l+M) 2.
A surprising result is illustrated by taking the limit M + 0 of the
end correction, giving CJ, which turns out to be different from that of
the no flow case (see (4,4.25)). As the value of w is restricted, this
limit yields incompressible flow. Only when M + 0 is taken first with
k = w M fixed, and then k + 0 , we would have the no flow case, The
double limit (M,k) + (O,O) is not uniform, which can also be seen by
comparing x (4.4.3) with x (4.4.8) for M = 0 and finite k . The
present problem is thus seen to provide an example of a non trivial
interaction of incompressible flow (at first sight acoustically
equivalent to stagnant flow) with sound waves.
80
An interesting feature, revealed by (4.4.22), is that on the interval
0 ~ z ~ 0(1/w) no instability wave can be observed by pressure measure-
ments. Almost all the flow energy is merely being convected. The
instability wave must be considered as a moving velocity - (vorticity)
distribution, fixed in the moving frame of the jetflow, comparable with the
undulating wake behind a flat plate, discussed in chapter 2. From
Bernoulli's law, we can
lity wave has the form
pressure p = - $t - $x
expect in our case, where the increasing instabi' $ "' exp [iwt-iw (1 +i ! 12 w log~ (l/w)) z] • that the
is negligible on the interval 0 < z < O(log!(l/w)).
The bonus, however, is an interference of the increasing.and decreasing
Helmholtz instability waves (corresponding to u 0
and u1), resulting in
a cancellation of the corresponding pressure even on the interval
0 ~ z ~ 0(1/w)
The vanishing of the pressure perturbations in the jet flow on the interval
1 << z ~ 0(1/w) (see (4.4.22)) can also be seen in the following context.
When wè compa.re the energy fluxes of the acoustic and hydrodynamic
disturbances in the flow far upstream in.side, and downstream outside the
pipe (Goldstein (1976), p. 41), we see that these are equal for vanishing
Strouhal number. Noting'that the energy in the jet flow is mainly kinetic
(compare (4.4. 16) with (4.4.22) for z>O), we can conclude that almost all
of the incident acoustic energy is transferred into the kinetic energy of
the shed vortices. This energy conversion is essentially the mechanism,
which Howe (1978) showed to be responsible for the attenuation of the
radiated sound power in the low frequency range, observed by Bechert,
Michel & Pfizenmaier (1977).
The important constant defined ( .-) Jo,n
(using Watson (1966) table VII,p. 748 , and
careful summation of the quantities
page 506), to find
c3
0.25538± 0.00001
An integral representation is
lo~[!xI0 (x)/I1 (x)] d~, x 0
to be compared with the representation
by (4.4. IS), is found by -1
- j 1,n) on a pocket calculator
the asymptotic expression at
(4. 4. 26)
1
J 0
1 dx log (2K
1 (x) I 1 (x)_ 2
x 0.6133,
which defines the end-correction in the classical no-flow case at low
81
Helmholtz numbers. The rather rough approximation (Watson (1966),p.506)
. (n - .!:.) 11 Jo,n 4
(4. 4. 27)
j (n + .!:.) 11 ' l ,n 4
only correct for large n, gives the promising re sult CJ = 4/11 - 1 = 0.273 ,
which is only 7% off. Using (4.4.27) in l: and l: 1 0 gives US the
opportunity to express the solution on the axis (r = 0) in quore or less)
elementary functions. We obtain then
~{-exp(-11z'),2,i (4.4.28)
+ 2 log 2 11 2 2
cosh(j 11 z') - /2
cosh(l 11z') + /2
4 arctg{ /2 sinh(l n z')} , TI2
i:1
(0,z') 2 1 12 5
<!> { - exp ( TI z') , 5 2 - ex:i(
4nz') 2.4 TI 112
812 exp(j TI z') 2
log 2 cosh(jnz') - /2
+ -- + TI2 TI2 2 cosh(jnz') + /2
4 arctg{/2 sinh(l 11 z')} , (4.4.29)
where <!> is Lerch's transcendent (Gradshteyn & Ryzhik (1965) ,(9.55)),
defined by n
q,(z, s, v) I _z_s n=O(n+v)'
( 1z1 < 1' v 1 0, -1' ... ) .
In appendix 6.2 we give sorne properties of
exp (vy) <J>{-exp(y),2,v} •
82
Using (6.2.2) we obtain, within approximation (4.4.27),
for 1z1 « (both z ;o 0 and z .:: 0 )
ik wk - 1l(i+M) - 1l(l+M)S (0.12 + 0.27M + (0.33 - M2) ~ + O(z2)] ,(4.4.30)
p(O,z) 110 ~~)S [o. 39 - 0.67 ~ + O(z2)] • (4.4.31)
Usually, the amplitude of pressure perturbations is presented in decibels.
Near z = 0 we have from (4.4.31) ( * quantities are dimensional ,
p;ef is some reference level)
_L _ 67 2z* 30 z* 20 log10 * - const. - 20 log 10 (e)39 öj) = const. - 0 i)•
Pref " P
(4.4.32)
Observe that the slope is negative. Here, near the pipe exit, we have only
the decreasing amplitude of the diffracted wave, because in the pressure
the instability waves are effectively absent, as we noted before.
Finally we will treat the far field outside the jet. The expressions
for x± we have now allow us to calculate the far field in a relatively
simple way. If we ignore the bow wave and the instability wave (which are
irrelevant far downstream, and would need much additional calculation) we
have, with the aid of the method of stationary phase applied to (4.3.14),
following the usual steps (Noble (1958)), that
1
(1 - M cosi';) 2 ik exp (-ikr) 411.
r
for kr ~ oo ,where z = r cos ~ , r = r sin ~,and O < ~ < 11 • To
(4.4.33)
make possible a proper comparison with the no-flow case, we have here
normalized the amplitude of the incident wave so that the incident wave
power is independent of M (i.e. instead of ~o we have taken (l+M) ~ 0 ),
From Noble (1958, p.115,138) we can compare with the classica! no-flow
solution (where the same power is incident along the pipe)
83
exp(-ikr) for kr ~ oo
r
The difference is clearly just a squared Doppler factor, one of which is
associated with the very slowly growing instability wave and the ether with
the decaying, When no vorticity is shed, we would have only the decaying
wave and thus only one Doppler factor.
4,5 Extensions downstream
The aim of this section is to extend the results inside the jet
downstream to z > 0(1/w) • Restriction (4.4.1) of z ~ 0(1/w) was -1
necessary when we employed the simple approximations u0
~ M and
u1
~ M-l , Obviously, we could have had a larger region of validity of
the previous solutions right from the start by using better approximations
However, this would have increased the complexity of the
formulae very much, while the present solution yields virtually all the
important conclusions already. Therefore it improves the clarity to defer
a refinement of the approximations to a later section, This is even more
worthwhile, because in the region z >, 0(1/w) the solution behaves much
more simply than further upstream, and so we only have to consider this
region separately. To see this, note the following: there is no primary
wave outside the pipe, the diffracted wave has an order-one exponential
decrease, so we have for very large z only the u0-instability wave left
over, and for z >, 0(1/w) the u0
- and u1-waves. Therefore it suffices,
to complete the analysis for all z , to calculate the contributions from
the poles u 0
and
For this we go back to.section 3,5,
easily deduced that
Using (3.5.4) it is
84
[X+ (u) J (u-u )
o u=u 0
(4.5.1)
- ( ) d ( ) (1 + O(w21ool)). X ul du X ul "'w (4.5.2)
Introducing these into (4.4.13) and (4.4.20) (i.e. $z and p) together
with (4.4.11) and (4.4.12) (i.e. and
possible, the relevant approximation, yields
x_), and applying, where
(note: x'= ~u x
(4.5.3)
[
(u -!)v(u ) o M o . • exp(-1ku z)
u2 x'(u) 0
1 (ul-M)v(ul)
0 0 uf x'(ul)
p(r,z) (4.5.4)
for z >, 0(1/w) and r ~ 1 The connections between the expressions
va lid on z ~ o(l/ w) with the expressions valid on z >, o(l/w) should
be understood as "matching". The range in z we can cover now depends
completely upon the accuracy we have in the approximation for u0
and,
until the decreasing u1
-wave is small, for u1
. So in principle we cai
reach every z . This is not necessary,however, because the thickening
of the real mixing layer limits the applicability of our model, and, after
all, our (linear) model is definitely not valid anymore in the region
where the waving vertex sheet reaches the axis r = 0 .
An example of an approximation for
z-region up to z ~ O(w-2 log!(l/w)) ,
u 0
is
and u1
which extends the
(use (4.4.4))
X (u ) 0
x r l <1 + ··M
(4.5.5)
log(! S ~xpy) + 2 w. log!(~) t -~ {.l.. + 2 log(j S expy)}.j 0 log (-) 0 log 2 cl) S2
w w
with Im t0
> 0 , and
v(ul)w2 1 [ -- -log(-) 2 t 2 + l w(u
1) k w l
(4.5.6)
'ITi+log(jSexpy) l 1 w tl f 1 . }} - 1 + 2wlog C;;;lt 1 ---1 -1~-+2n + 2log(jBexpy)
log C;;;) log 2 <;;;l s 2 0
with Tm t1
< 0 .
Equations (4.5.5) and (4.5.6) give, via a simple algebraic equation, u 1 0
O(w 3log 2 (l/w)) and and u1
with relative errors of
O(w3 log 312 (1/w)) respectively. Moreover, they provide estimates for d du x at u
0 and u1 . Differentiation of (4.5.5,6), and then, finally,
85
substitution in (4.5.3,4) of all the results obtained, yields, respectively,
to leading order
~u x(uo) - ~u x(ul) ~ 2/2 is logi(;)
ij>z(r,z) ~ 2'IT(~~M) [ exp(-iku0
z) + exp(-iku 1z) J , 1
Mw2log2
(1/w) [ J p(r,z) 2/2'IT(l+M) exp(-iku0
) - exp(-iku 1)
(4. 5. 7)
(4. 5. 8)
(4.5.9)
And hence, we obtain for large z (using the same notation as in (4.4.32))
_L- _ 2z* 2 1 1 z* 20 log 10 * - const. + 20 log 10 (e) Im(u
0) k -n- ~ c. + 12.3 w log 2 (-w) D .
Pref (4.5.10)
86
±!.§.. Comparison with ewperiments
As a general remark it should be noted that it is difficult to
compare the calculated sound pressure levels with the measured ones.
Owing to the different resonance characteristics of a real pipe system for
different Mach numbers, we have unknown effective incident wave amplitudes
to scale on. Therefore we shall only consider the variations of the
measured data.
As we pointed out before, the present model was evaluated numerically
by Munt (1977) for the far field, and the agreement with the experiments
of Pinker & Bryce (1976) was very good, We shall compare our results with
Munt's; agreement, if found, implies then innnediately agreement with the
experiments. The far field expression (4.4.33) is, normalized, in
decibels
const. - 40 log 10 (1 - M cos(E;)). (4.6.1)
Representative values obtained from this expression are compared with those
of Munt (Fig. 13 and 14 of Munt) in the following table,
{SPL(900) - SPL(l70D)} {SPL(500) - SPL(l70D)} w 'M Munt, expr.(4.6.1), rel.error Munt, expr.(4.6.1), rel.error
0,27 0.9 10.4 11.0 6 % 24.5 26.0 6 % 0.40 0.6 7.6 8.1 7 % 15.8 16.5 4 % 0.67 0.9 10.1 11.0 9 % 22.7 26.0 15 % 0.80 0.3 4. 1 4.5 10 % 7.8 8.2 5 % 1.00 0.6 7.0 8.1 16 % 15.0 16.5 10 % 2.00 0.3 3.0 4.5 50 % 7. 1 8.2 15 %
where {SPL(;) - SPL(l70D)} denotes the increase of the sound pressure
level in decibels from the value at 1700 to that at ; It is seen that
the present results agree well with those of Munt. Thus the simple
expression (4.6.1) is capable of representing the far field for Strouhal
numbers w right up to a value of one, possibly even two. Another
feature of these calculations is that it is easy to observe the role that
Kutta condition plays in the far field. When this condition is absent, the
factor 40 in (4.6.1) would be 20. Therefore the agreement with the
experiments shows that vortex shedding contributes signific'antly to the
far field.
87
Experiments relevant to the present results inside the flow are
provided by Moore (1977). The expressions for the sound pressure level on
the jet axis (4.4.32) and (4.5.10) show features which are in
qualitative agreement with the observations presented in Fig. 20 of
Moore's paper. The slope near the origin, predicted to be -30/8 , is of
the right order of magnitude, which also confirms the effective absence of
the instability waves immediately downstream from the exit. The positive
slope corresponding to the instability wave further downstream from
the experiments for Strouhal numbers sufficiently below unity is of the
same order of magnitude as that predicted by (4.5.10) ( 12.3 w2log!(l/w)).
We see that in the experiments without flow the behaviour of the pressure
outside the pipe is of completely different character than generally is the
case in the presence of flow, viz. a monotonie decrease. For M sufficiently
small and k not small ( M = O. 15 , k = 0.5, and M = 0.3 , k = 1 .2 ) an
intermediate state is seen, which may be a reflection of the non uniform
limit behaviour when (M,k) + (O,O).
The lack of quantitative agreement is believed to be mainly due to the
finite length of the tube used in the experiments. This causes resonance
effects at low frequencies as can be seen in Moore's figures. Moreover,
the smallest Strouhal number w in these experiments is only 0,5 ,
which may be too large to allow detailed comparison.
4.7 Conclusions
We have calculated for a circular subsonic jet, issuing from a pipe
and perturbed by long sound waves, the axial velocity and pressure inside
the flow, and the far field pressure outside. The Kutta condition is always
applied, except, for the purpose of comparison, in a single case, As in
chapter 3 , also here a causality condition is superfluous. By comparing
with the solution without vortex shedding, the Kutta condition appears to
affect the magnitude of the reflection coefficient, hut not the end
correction. In the far field the regular solution differs from the
singular one in having a squared Doppler factor instead of one. When no
vorticity is shed, practically no power is transmitted out of the tube, and
when the Kutta condition is applied, there is a power loss of O(M),
The limit (M,k) + (0,0) appears to be non-uniform, When M + 0 with
88
k = wM fixed, and then w + O we obtain the no-flow case: the end
correction is then t = 0.6133 , while the sound does not feel the flow
anymore and spreads from the jet mouth as from a point source. On the
other hand, when w + 0 and then M + 0 , 'we have an incompressible flow,
for which case the interaction between sound and the flow is retained; the
end correction is now CJ = 0.2554 , This is regardless of the application
of the Kutta condition, The interaction of the increasing and decreasing
Helmholtz instability waves (when the Kutta condition is satisfied)
results in a relatively long interval along the jet axis, where (1) the
velocity perturbations of the instability waves are purely harmonie, and
(2) the corresponding pressure perturbations are negligible, Hany aspects
agree quite well with the experiments.
Note adàed in proof
Prof.dr. J, Boersma, after going through the manuscript, suggested
a more direct way of deriving the end corrections t and acJ (see
(4.4.25)), which at the same time clarifies the non uniform double limit
behaviour for (M,k) + (0,0),
Noting that the reflected wave corresponds to the residue in the pole
u = -1/(1-M) of the relevant integrand, the exact reflection coefficient
is, from expression (4.4 •. 20) for p(r,z) , readily found to be
1 R = - k
1 (4-M2) ! X_ (J+M) --- -1 (l-M)2 X+(l-M)
After careful investigation of
representations of the split functions, of the kind given by (3.5.1),
this expression for R can be approximated for small k , uniform in M , '
as R • - exp(-2ikL/S2) , where L , the end correction, is given by
L = l f"' log{ 1T 0
I (St) !at - 0
--
11 (13t)
K (t) dt _o __ }
1<1(t) t 2
Using (4.4.5) for
when M + O, and the identity I0K1+I
1K
0=1/t for when M=O , the two forms
acJ and J1; follow immediately.
chapter 5
ACOUSTIC WAVES INTERACTING WITH VISCOUS FLOW NEAR AN
EDGE: APPLICATION OF RESULTS FROM THE TRIPLE DECK THEORY,
5.1 Introduction
89
The acoustic problems, studied until naw, were all simplified in their
neglect of viscosity. In this chapter we shall discuss the consequences
of the introduction of viscosity, especially with respect to the Kutta
condition. Obviously, it must be kept in mind that the inviscid models
of the previous chapters are supposed to apply "almost", i.e. to be
approximations, Therefore the influence of viscosity will be taken as
small. This will imply that we shall ignore the internal friction in a
sound wave, and only consider the boundary layers near the solid body.
Note that the sound wave farms a boundary layer near the wall (the Stokes
layer) as well as the main flow (the Prandtl- boundary layer), and they
may interact. In the end, however, we shall find that only the regions of
the boundary layers near the trailing edge are crucial for tre flow and sound
field interaction, and only these regions are therefore interesting for us.
It is tacitly assumed that the flow is laminar, We try to deal with
rational theories only, and there does not seem to exist any such rational
theory for turbulent flow.
Let us, as a start, confine ourselves to a consideration of a flat
plate of length L , aligned parallel to an incompressible uniform flow
with velocity U and kinematic viscosity v , and with large Reynolds
number Re = UL/v Then it is well-known that along the plate a thin
boundary layer develops (the Basius boundary layer; see for instance Van
Dyke (1975)), eventually giving rise to a wake bebind the plate (the
Goldstein wake; Goldstein (1930)), According to Stewartson (1969) and
Messiter (1970), it appears that the transition from the boundary layer
to the wake is quite a delicate matter. Because of the sudden change in
boundary conditions at· the trailing edge - from zero tangential velocity to
zero stress - the flow in the boundary layer accelerates, which reduces the
90
boundary layer growth, This in its turn causes a small hut sudden
deflection of the main flow streamlines, which induces a singularity of
the mainstream pressure at the trailing edge, This singularity is
smoothed out in a three-layered boundary layer structure (the "triple
deck") around the trailing edge, of dimensionless proportions
0(€3) x 0(E 3) , where Re-l = E8 and E is small. Inside the triple
deck the dimensionless pressure is of order E2 (which is higher than
the surrounding boundary layer-induced pressure perturbations of order
E4 ). This triple deck pressure is a result of the smoothing process. So
an externally generated disturbance will only change this process ,andthus
possibly change the flow, when its corresponding pressure at a distance
from the edge of order E3 is of the same order as the triple deck
pressure (0(E 2)), or larger. We shall, however, not consider the case
of a pressure disturbance much larger than 0(E 2) , because in that case
separation probably starts, for which as yet no theory exists. Although
a pressure disturbance of 0(€2) does modify the triple deck structure,
the resulting outer field is only slightly different from the completely
smooth (Kutta condition) field. Therefore we can conclude that for this
class of disturbances the smooth outer solution adopted in previous
chapters is valid on a rational basis. All these observations,
essentially supplying a rational basis for the smoothing processes at the
trailing edge and leading to an edge singularity condition for the
inviscid outer solution, are generalizations of several deeply studies
examples, We have, for instance, the problem of a flat plate at
incidence in steady incompressible flow (Brown & Stewartson (1970)) and
in steady supersonic flow (Daniels (1974)), a rapidly oscillating flat
plate (Brown & Daniels (1975)), and the problems of wedge and cusp shaped
trailing edges (Riley & Stewartson (1969)), although these last two fall
slightly outside the scope of the form of Kutta condition relevant here,
In addition to these more or less obvious variants of the flat plate
trailing edge problem, the triple deck notion appears to provide an
understanding of many separation problems of all kinds (see for instance
Smith (1977)); indeed its first appearance was in fact in a study of
self-induced separation of supersonic flow (Stewartson & Williams (1969)),
In view of the problems studied in chapters 3 and 4 of this thesis, our
special interest is in the separation behind a flat plate trailing edge,
where a uniform flow together with a Blasius boundary layer on the one
91
side and a stagnant flow on the other form a mixing layer behind the plate
edge (Daniels (1977)). A peculiar property here, is that in the
undisturbed case the triple deck structure disappears (in the main order
of the approximation), because the flow can find an easier way to deal
with the pressure singularity. The deflection of the main flow stream-
lines near the trailing edge - the result of the acceleration of the
boundary layer flow behind the plate - is not necessary now. Prandtl's
transposition theorem for the Prandtl boundary layer equations (Rosenhead
(1963,p.211)) implies the possibility of a small shift of the dividing
rreamline (the streamline emanating from the trailing edge) in the
Jirection of the main flow, which suffices to compensate for the
acceleration. This results in an outer flow singularity of only minor
importanc.. In analogy with the airfoil, it is evidently to be expected,
and indeed the case, that an external disturbance, corresponding to a
non dimensional pressure of 0(E 2 ) in the triple deck region, calls the
triple deck structure (modified on the stagnant flow side of course) back
into play. Daniels (1978) studied this case for the Kutta-condition-
outer solution of Orszag & Crow (1970). Although, as Bechert & Michel
(1975) showed, this outer solution (as well as Orszag & Crow's so-called
"rectified Kutta condition"-solution) is not an eigensolution and only
valid with the presence of a very unusual source, Daniels' (1978) viscous
solution is very useful and generally valid, because every other smooth
outer solution behaves, modulo a factor, in the same way in the triple
deck region.
In addition to his calculations concerning disturbance pressure levels
of order c 2 in the triple deck, Daniels found the possibility of a "no
Kutta condition" outer solution in the mixing layer problem with stagnant
flow on the one side. When the perturbation pressure in the triple deck
is sufficiently small, i.e. of order E912
, he showed that the singularity
of the no vorticity shedding solution is not inconsistent with the triple
deck dynamics, and may be smoothed out in the Navier-Stokes region of
dimensions 0(E6) x 0(E6) around the edge. However, as Daniels argues
on more general grounds, the whole point may be irrelevant altogether, and
furthermore the case of even smaller perturbations is &till an open
question, so that we shall, for the present, only note it, and not include
it in our acoustic problems.
92
The aim of the present chapter is to introduce the triple decks into
the acoustic problems of chapters 2 and 3, to prove, for a certain range of
parameters, the validity of the Kutta condition in laminar flow of high
Reynolds number. Further, for some limiting cases, we shall give explicit
expressions for the far-field correction, due to viscosity, to the
inviscid full Kutta condition solution. Although it is not essential
(Brown & Stewartson (1970,p.584)) we shall restrict ourselves to a Mach
number small enough to give in the triple deck incompressible flow to
leading order. Furthermore, we shall choose the Strouhal number
suffîciently high that the Stokes layer has dimensionless thickness 0(E 5),
which is the thickness of the lower deck in the triple deck. These
choices are only for convenience; they permit analytically tractable
problems, and moreover, they allow us to practically repeat the existing
calculations, namely those of Brown & Daniels (1975) for chapter 2 and
those of Daniels (1978) for chapter 3.
Although virtually impossible to prove, it is most likely that order
of magnitude estimates themselves already inform us also about the
validity of the Kutta condition in the cases we will not consider. So we
postulate that when the dimensionless pressure of the smooth outer solution
behaves like p = constant + A( w,M) /r exp (iwt) + O(r) (r ->- 0), the Kutta
condition is valid (in laminar flow of high Reynolds number) when
A(w,M) = O(~) = O(Re-1116), (the constant can be ignored as it does not
contribute to the gradient).. It should be noted that in reality, a
Reynolds number E-S so high that even lË is small, is impossible in a
harmonically disturbed laminar flow. A Blasius boundary layer is
urtstable for Reynolds numbers beyond 6.10 4 , and definitely turbulent for
Reynolds numbers higher than 3.106 (Rinze (1975)). This may even
invalidate the whole triple deck concept for practical applications.
However, there are reasons - besides the better understanding of the
mechanisms - to relax the "high Reynolds number" condition and to give the
asymptotic expressions from triple deck theories at least the benefit of
the doubt. The skin friction coefficient of a flat plate, based solely
on the contributions from the Blasius boundary layer and the triple deck,
appears to coincide with measurements and numerical calculations in an
astonishing way right down to Re = 1 (Jobe & Burggraf (1974) or Veldman
(1976))and is still very good for Re
solution with Dennis & Dunwoody (1966)À
0,1 (compare Veldman's (1976)
As a second example Daniels (1978)
unsteady solution is in good qualitative agreement with the measurements
of Bechert & Pfizenmaier (1975).
We hope to have given a general idea of the relevant aspects of the
triple deck and its modifications, at least sufficient to enable the
reader to follow the applications to our acoustic problems in the next
sections. We shall not consider the different triple deck layers and
subboundary layers in detail. We do not claim to contribute anything
important to the triple deck theory itself, so we think that for those who
are interested in its detailed description, the original literature is
most recommended.
5.2 Formulation of the problems
In the chapters 2 and 3 we considered, for convenience and clarity,
a semi-infinite plate. Obviously we would like to preserve this geometry
in the present viscous problems. However, a careless introduction of
viscosity would rule out the possibility of an inviscid outer solution,
because the influence of the plate would then be felt everywhere,
Therefore, we shall modify the properties of the plate a little with the
rather artificial assumption that the plate is frictionless everywhere,
except for a finite part upstream of and including the trailing edge. The
dimensional length of this part is L; it will be our reference length-
scale when we make the problem non-dimensional. The coordinate axes are
93
chosen in the same way as in chapters 2 and 3 (see figures 1 and 2). We
assume, in the whole medium, the validity of the two-dimensional time
dependent compressible Navier-Stokes equations, the compressible continuity
equation and the energy equation, together with the appropriate thermo-
dynamic equations of state of a gas (Batchelor (1967). We make the
coordinates x*, y*, t*, the velocities u* and v*, the pressure p*, the
density p* and the temperature T* dimensionless in the following way:
x* xL, y* = yL,
p* T*
t* = tL/U,
= T0
(l+T),
u* = Uu, v* = Uu, p* = p0
+ p0
U2 p,
where u is the main flow velocity and Po (l+p)'
and T po, po 0
are, respectively, the density, pressure and temperature
at infinity. Introduce the Reynolds number Re
the viscosity coefficient, and the Mach number M =
sound speed at infinity. For convenience we write
p0
UL/µ where
U/c where 0 -1
Re = EB
c 0
µ is
is the
We
assume the Reynolds number large (so that E is small), the Mach number
94
small (M = E2M ) , the dimensionless frequency, or Strouhal number, of 0
the acoustic disturbances large (w = E~2 w so that k = wM o'
and the plate temperature equal to the flow temperature. The second
viscosity coefficient µ' is of the same order of magnitude as µ, i.e.
µ'/µ = 0(1). When we assume that the perfect gas relation is sufficient
to provide order, of magnitude estimates, then we can deduce that the
assumption IPI ~ O(/Ë) (in facta result of the subsequent theory)
reduces the energy equation to the condition of isentropy, with the
solution T = (y-l)M2p (where y is the ratio of the specific heats).
Furthermore, it follows that the pressure, density, sound speed,
temperature and viscosity coefficients are, to the accuracy considered,
constant (but not necessarily their derivatives of course). Thus, for
example, in the continuity equation (l+p)ux is taken as u while x
u(l+p)x we retain upx • We shall adopt this form from the start as
eases the calculations very much. From the isentropy and the constant
sound speed it follows that p = M2p , the acoustic approximation.
in
it
After substitution of the above, we have the equations and boundary
conditions (in which the suffix x means differentiation w.r.t.
similarly for y and t
M2(p + up +vp ) + u + v 0 ' t x y x y
+ uu Es [u + µ' v ) ] ut + vu + px u + -(u +
x y XX yy µ XX xy
+ + vv + py Es [v + µ' + v )] vt uv v + µ<uxy ' x y XX YY YY
u(-1 i; x i; 0,0) 0 '
v(x ~ 0,0) 0 •
x . '
(5.2.1)
In the inviscid outerflow (roughly speaking: when x,y ~ 0(1)) the field
is given by that of chapter 2 (in section 5.3), and that of chapter 3 (in
section 5,4),
95
5.3 Harmonie plane waves with a wake
As we saw in the introduction 5.1, the most important variable in the
triple deck is the pressure. The appropriate expression f or the pressure
(now normalised) of the plane wave problem is found from equations (2.4.3)
and (2.3.4) as
p a(cjl + B pe. exp(iwt)), c,s 0
(5.3.1)
with pe (x,y,w) sin je
exp (-iKR + iKMX), k SK
where the amplitude a is to be chosen in such a way that it gives rise to
0(E 2 ) pressure perturbations in the triple deck region. It is clear
from equations (2.4.2) or (2.4.4) that when the waves carne directly
from downstream, SO that ei = eS 08
= TI and cosj0S = Ü , the
continuous solution is smooth already, and no vorticity will be shed.
Therefore, when ei ~ TI considerable differences may occur in the way the
triple deck is perturbed. Such cases are probably all quite tractable,
but have only marginal importance;
and shall adopt now the restriction
therefore we shall not consider them
TI - 8. >> E312 •
1
In the triple deck region r = E3r3
x = E3x3
, y = E3y3
r 3 ,x3
,y3
= 0(1) , we have after application of all the relevant
approximations and neglect of the unimportant constant term
p
1
- E2a0
2 (2nk)
2 cos(lei)sin(je) exp(if + iwt) [rr
3
where B = -E3 2 0
that makes p
l 2k 2
TI (-:;;-) cos Oei) exp(i"4) B1 , and a =
0(E 2). This is the analogous form
(5.3.2)
l E
2 a the choice 0
for the pressure to
that of Brown & Daniels (1975) given by their equation (2.7). We checked
the correspondence in all detail, and it appears that the disturbances of
the steady flow boundary layer structure are in essence the same as in the
incompressible problem of Brown & Daniels. In the region r = 0(E 2) the
disturbances are already incompressible at leading order. Outside this
region, external disturbances simply govern the disturbances in the
boundary layer, without being affected by any interaction, while the limit
behaviour near the edge is the same as in Brown & Daniels. Theref ore
there is no need to repeat all the calculations, and we can at once simply
96
use Brown & Daniels' principal result, namely an estimate of B 0
for
w >> 1 (obtained o -11
by linearisation of the lower deck problem, which implies
a (Mw f ->- 0). 0 0 0
After identifying our parameters in their equation
(7. 32), we find
B 0
where À = o.3321
(5.3.3)
a constant arising in Blasius' boundary layer !
solution. Note that (Àrr) 2 = 1.0215, and that the linearisatión is valid
for all fixed a ,M as w ->- oo • In the far field this correction is 0 0 0
felt in the following way. Using (6.1.3) to find the asymptotic
behaviour of the diffracted part of the pressure wave, we have
! sin(j0)cos00s) e: a ----~--;:;__ exp(-iKR + iKMX + iwt) •
0 (rrKR) ~
[- 1 . w !] 1
- i + e:3M (....E.) • (1 +0(-)) cos0 - cose o À R
s (R ->- oo) • (5.3.4)
Clearly the viscous interaction at the trailing edge yields only a
small correction to the leading order outer solution (the Kutta condition
solution), and hence we can conclude that the assumption of the Kutta
condition is consistent with triple deck structure. Nevertheless,
experimental verification of the extra term in (5.3.4) might perhaps be
possible, and would constitute an interesting indirect confirmation of the
triple-deck theory, whose direct experimental verification is difficult.
The behaviour of the diffracted wave of the harmonie point source
problem (~ in section 2.3, or Jones (1972)), is essentially the same as
given here by (5.3.3), provided the source is not inside the triple deck
region (say, ro ~ o<i>>. The only difference is that the amplitude a
is now a function of the source position, and 0s should be replaced by
The calculations, yielding the precise form of a , are 0 = 0 - rr • 0 s
quite tedious, do not reveal anything new, and moreover, the results are
of much less practical value than those of the plane waves, because of the
more difficult determination of the wave amplitude. We shall therefore
not consider the pointsource problem here in detail, and end with the
remarks that in Brown & Daniels (1975) , equation (4.8) suffers from
numerous misprints, while expression (4.3) does not match with the upper
97
deck expression (5.9). In (5.9) an 3/2 O(E ) term is missing, but it
appears to be onl.y a constant which does not imply any further change.
5.4 Harmonie plane waves with a mixing Zayer
As was explained in the introduction 5.1, the main lines of reasoning
in the unsteady mixing layer problem are the same as those in the unsteady
wake problem. Here we shall use Daniels' (1978) incompressible viscous
solution as the inner solution to match our compressible inviscid solution
found in chapter 3. The relevant variable is the pressure on the flow
side, expressed by equations (3.4.21) and (3.4.23), or, approximated, by
(3.6.17) and (3.6.19). After application of (6.1.2), we find for the
normalized pressure in the triple deck region near the plate (y = -0,
x < 0)
constants +
where
k j [ 8
1 J E2 a 2(l+i) (-) ~ + --0 'IT 3 ~
and B 0
3
exp(iwt) + ••• , (5,4.1)
Daniels (1978) linearizes the fundamental lower-deck problem in the limit
of small amplitude, and finds an estimate of B1
for w >> 1 (in our 0
terminology). Ris expression (4.9)
p 0, -9,
where h is the amplitude,
+ ••• } '
S the Strouhal number, the constant
À = 0.3321 and his x2
is our x3 , is essentially the same as our
equation (5.4.1), provided we take the complex conjugates,as Daniels' time
dependence is exp(-iwt). Identification of the different terms yields
98
s IJ) +""
) 0
exp (-i2.11) M h* 0 8
+ 0 ' a 25/4 0 IJ)
0
lrl B E [M0 (w0À)
2
0
Note that the condition h+O is automatically satisf ied when
with the other parameters fixed,
The most interesting far field is in the still region, with
(5,4.2)
(5.4.3)
IJ) +"" 0
1T e < 1T • 4 < Taking into account the above viscous correction we have
there (cf. (3.6.20); use (3.6,16) and (6.1.3))
! ! E 2 a .!_ (~)
0 4 1T
2w0 ! ] - E3M (~-) sinje exp(iwt) + ,,,
0 À -(kr + 00) (5.4.4)
We see that the contribution of the viscous correction is even smaller
than the sound associated with the vortex shedding, while both are smaller
than the error terms inherent in equation (3.6.14), the approximated
version of (3.4,18), Obviously a better approximation of (3.4.18) is
needed, which is not readily to hand, however, Therefore we have to
conclude that as yet the use of expression (5,4.4) is limited to showing
when and how deviations occur from the full Kutta condition state, and of
what kind these deviations are.
We finally note that Hautus & Brands (1976) have proved the
existence and uniqueness of the most important function in the problem of
the mixing layer near the edge (Daniels (1977)), It is the function f0
,
def ined by f " ' + ~ f f " - l f ' 2 o f ' "' À n + o as n + "" , o 300 3o o
f0
' + 0 as n + - "" (equations (2.9) and (4.11) in Daniels (1977)), The
non-linear character of the equation made it far from obvious whether the
boundary conditions were possible,compatible, superfluous or sufficient,
It turned out that they were possible, compatible, and just sufficient.
5.5 Conolusions
We have discussed the mechanism of the viscous smoothing process at
the trailing edge in our acoustic problems of chapters 2 and 3, resulting
in the application of the Kutta condition to the inviscid outer solution.
Making use of the work of Brown & Daniels (1975) and Daniels (1978), we
derived an outer field correction, due to viscous action at the edge, for
high Reynolds numbers. It appears that an incident pressure wave with a
dimensionless amplitude of order /€ andfrequency of order E-z almost
satisfies the Kutta condition;
with an amplitude of order E 3 a multiple of the singular eigensolution
is to be added. It is conjectured that
when a pressure wave, satisfying the Kutta condition, behaves near the
edge like p = const. + A(w,M)/r exp(iwt), where A = O(IË), the
Kutta condition will, indeed, have a rational basis.
99
100
chapter 6
APPENDICES
6.1 Proper>ties of Fr>esneZ's integr>aZ F
definition
F(z) exp(iz2) f exp(-it2) dt
z
series expansion
(6.1.1)
F(z) Îl7i exp(-iÎ)(l + i z2 -Ïz" -ii z6 + ".)
- z ( 1 + ~ i' z2 - ..!_ z" 8 · 6 ) 3 15 - 105 l z + •••
l (i z2)n {....!.. .!. l7i exp(- i.'!!.) n=O n! 2 4
asymptotic behaviour
- 22n n! z}
(2n+l) !
(z -+ 0) = (6.1.2)
(-co< z <co),
F(z) i 1 1 - - + - + 0(-)
2z 4z3 z5 , 1 z 1 -+ co in -TI < arg z < I ,
i 1 1 l7i exp(-i-'+47T iz2) - - + - + 0(-), lzl-+ co in f~arg z ~ 1T.
2z 4z3 z5
(6.1.3)
functional relations
F(z) + F(-z) /7i exp(-iÎ + i z2), (6.1.4)
d dz F(z) 2 i z F(z) - 1 , (6.1.5)
101
exp(- i z2) r exp(i t 2) dt i F(-iz) (6.1.6) J z
We note that the function F has been tabulated for complex z by
Clemmow & Munford (1952) and Rankin (1949).
In flat plate diffraction problems, F is aften met with the argument
(2kr)! sinj(e ± e ). Therefore it is useful to complete this section 0
with notes on this argument in particular.
De fine y (2kr)! sin! (e - e ) 0
y (2kr)! sin! (e + e ) 0
then we have (index x{y) denotes differentiation w.r.t. x(y))
1 1 -
- 2r y (~)' 1 ( ) 2r sin 2 e + e 0
k l <2r) cos l (e + e
0)
l k (case - case 0
)
l k (sine - sine 0
)
2k l . - <-;-> sm l e cos l e 0
2k l 1 <-;-> cos 2 e cos l e 0
(2k) l cos l e · f ~ exp (-ikr + ikx case )dx = o r2 o
(6.1.7)
(6.1. 8)
(6.1.9)
(6.1.10)
(6.1.11)
(6.1.12)
(6.1.13)
{F(y) + F{y)} exp (-ikr + ikx cos6 ) + { function of y }. 0
102
6.2 Properties of Lerah's transaendent ~
Re sul ts are given for ~ in the form in which we meet i.t in
chapter4, i.e. exp(vy) ~(-expy,2,v).
Directly from the def inition, we have (for - ® < y ' O )
exp(vy) ® (-l)n exp(ny) }:---
(n + v) 2 ~ ( -exp y , 2 , v) exp(vy)
n=O (6.2.1)
Applying Gradshteyn & Ryzhik (9.554), (9.531), (9.623.3) and (8.363.8)
yields
exp(vy) ~ (- exp y , 2, v)
for IYI < 11 , where
_ .P'(v+l)} + 2
1 (2y)n - B (~)} ~~_.,_,,__,~
n+l 2 (n+l) (n+2) ! (6.2.2)
Bn(x) is the n-th order Bernoulli polynomial according to the definition
of Gradshteyn & Ryzhik (9.62), and
iji(x) is the logarithmic derivative of Euler's gamma function (Gradshteyn
& Ryzhik (8.360)).
Some numerical values (Abramowitz & Stegun (1965)) are
ijl' (l) 8
- ijl' (2) 8
6.1560
ijl(l) 8
- ijl (2) 8
- /2 11 + /2 log (3 + 2 12) -1. 9500
ijl' (l) 8
- ijl' (2..) 8
64 - 4 .fi 11 2 - ijl' (l) 8
+ijl' <i) 2.0130
ijl(l) 8
- ijl (2..) 8
12 11 + /2 log (3 + 2 12) - 8 -1. 0642
3 - B (2) l 5 9 Bl (3) 1 8 - 2 Bl (3) Bl (3) -2
3 7 1 5 9 3 B2 (3) B2 (3) - 8 B2 (3) B2(8) 8
3 7 9 5 - B (2_) 15 B3 (3) B3 (3) ill B3 (3) . 3 8 - ill
3 - B (!_.) 11 5 - B (2_) 9 B4 (3) 4 8 256 B4 (3) 4 8 256
3 - B (!_.) 285 5 - B (2_) 275 BS (3) 5 8 - 8192 B5(3) 5 8 8192
For ,all y with - 00 < y < 0 , we find from Gradshteyn & Ryzhik (9.556)
that
103
exp ( vy) 1> ( - exp y , 2, v) 1 exp(vy)
1 - J ---...,-( y - - log t ) dt , (6.2.3) v 0 ! v
1 + tv
of which, for v=l±l , the term 1 exp(vy)
1 - f -- dt v 0 1
1 + tv
exp(!y) 3±1 4 J _x __ dx
0 1 + x4
can be expressed in elementary functions (Gradshteyn & Ryzhik (2.132)),
104
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109
110
LEVENSLOOP
10 maart 1952
1964 - 1970
1970 - juni 1975
augustus 1975 - juni 1979
NAWOORD
- geboren te Manado, Indonesië
- gymnasium B, Openbare Scholengemeenschap,
Utrecht
doctoraalstudie Wiskunde, Rijksuniversiteit
Utrecht; afstudeerrichting "toegepaste
analyse" bij prof.dr.ir. W. Eckhaus
wetenschappelijk medewerker aan de T.H.E.,
vakgroep Transportfysica
Allen die mij hebben geholpen bij de totstandkoming van dit
proefschrift wil ik hierbij hartelijk danken. In het bijzonder noem ik:
de leden van de vakgroep, speciaal de mede L.T.D.-ers, voor de
prettige samenwerking en de vriendschappelijke sfeer; bovendien nog apart
ir. L,J, Poldervaart die mij aanzette tot het leggen van de contacten
met Engeland,
de Universiteit van Leeds voor de gastvrijheid, en de medewerkers
van het Applied Mathematica Department aldaar voor de enthousiaste
ontvangst en belangstelling,
mrs P. Jowett en mevrouw H. Weisè voor het vlot en keurig
verzorgde typewerk, en Ruth Gruyters voor de nette tekeningen.
lil
SAMENVATTING
Dit proefschrift behandelt de interactie van geluid met een stroming. Er
worden drie modellen bestudeerd waarin steeds de aanwezigheid van een
afstroomrand een centraal element vormt. Een uniforme subsone wrijvingsloze
compressibele stroming aan één of beide zijden van een half oneindige dunne
vlakke plaat, en in een half oneindige dunwandige open pijp wordt verstoord
door geluid,
én het probleem met een stroming aan beide zijden van de plaat worden een
Jlindrische geluidspulsen een vlakke harmonische geluidsgolf verstrooid
door de rand en de stroming. Expliciete resultaten, exact binnen de
gebruike, '.jke linearisering, zijn verkregen voor het resulterende veld.
Afhankelijk van de mate van het bij de rand op te leggen singulier gedrag
van de oplossing, wordt aan de rand wervelsterkte geproduceerd. Deze wervels
maken zich akoestisch kenbaar in de diffractie golf. De reguliere oplossing,
met maximale wervelproduktie (de Kutta voorwaarde is dan geldig), heeft
stroomopwaarts een lagere en stroomafwaarts een hogere drukamplitude dan de
singuliere oplossing zonder wervelproduktie.
In het probleem met een stroming aan één zijde wordt een vlakke harmonische
geluidsgolf, komend van stroomopwaarts vanuit de stroming, weer verstrooid
aan de rand en genereert, afhankelijk van de op te leggen randsingulariteit,
wervelsterkte. We hebben nu echter achter de plaat een tangentiële
cnelheids-discontinuiteit (wervellaag) in de hoofdstroom. De hierin
inL. ·ente Helmholtz instabiliteit wordt aangeslagen indien wervels worden
gegenereerd, waardoor het hydrodynamische veld ingrijpend kan worden
gewijzigd. Exacte oplossingen in integraalvorm worden gepresenteerd,
waarbij wordt aangetoond dat het de conditie aan de plaatrand is die de
instabiliteit beheerst, en niet een zekere causaliteits eis, ingevoerd door
vroegere auteurs. Uit de exacte oplossingen worden expliciete benaderingen
verkregen, geldig voor kleine waarden van het Mach getal.
In het pijpprobleem komen vlakke harmonische geluidsgolven van stroom
opwaarts uit de pijp. Naast diffractie en de produktie van wervelsterkte
treedt nu ook reflectie aan het open pijp uiteinde op. Exacte oplossingen,
weer in integraalvorm, worden benaderd in de limiet van lange golven (klein
Strouhal getal) door expliciete uitdrukkingen. Onhafhankelijk van de
112
gegenereerde wervels, blijkt de "eind correctie" bij een incompressibele
stroming essentieel te verschillen van die bij geen stroming (t.w. 0,2554
en 0,6133 respectievelijk). Dit hangt samen met het niet uniform zijn van
de dubbele limiet "Mach getal-+ 0, Strouhal getal-+ O". Verder wordt in de
singuliere oplossing zonder wervel produktie vrijwel de gehele ingevoerde
akoestische energie aan het pijp einde gereflecteerd, terwijl in de
reguliere oplossing een deel van orde M (Mach getal) ontsnapt. Dit wordt
echter omgezet in hydrodynamische energie (wervels), en is vrijwel niet
merkbaar in het buiten gebied.
Tenslotte worden de eerste twee modellen uitgebreid met viscositeit om
enige rationele basis te geven aan de wervel produktie aan de rand. Bekende
resultaten uit de "triple deck" theorieën voor incompressibele stroming
worden ingepast. In een laminaire stroming, en met de waarden der parameters
beperkt tot een zeker gebied, wordt voor grote waarden van het Reynolds
getal de correctheid in eerste orde van de Kutta conditie aangetoond;
tevens wordt de visceuze correctie term gegeven.
-4
Addendum to chapter 4.
The waves, diffracted at the pipe exit, are given in
(4.4. 16,17,22,23) in terms of the functions Z0(r,z') and
(see (4.4. 18, 19)). These are approximated on the pipe axis
Z 1 (r,z')
r = 0 by
(4.4.28,29). After numerical integration of the respective expressions
(6.2.3), these approximations of zo,I (O,z') are (on logarithmic scale)
given by the following diagrams:
-3 -2 -1 0 2 3
z '-axis
-7
4
STELLINGEN
1. Beschouw het "forward-time centred-space" differentie schema voor de
numerieke oplossing van de tweedimensionale tijdsafhankelijke Navier
Stokes vergelijkingen.
De opvatting, zoals bv. geponeerd door Roache, dat de zg. "cell
Reynolds number" conditie voldoende en noodzakelijk is voor stabiliteit
van het schema, volgt niet uit berekeningen. Voor ingewikkelde
configuraties is het de vraag of de conditie is te bewijzen, en voor
eenvoudige is de conditie te verscherpen. Bijvoorbeeld wanneer de
stromingssnelheidsvector in het rekenpunt langs een der assen van het
rooster is gericht (zegge (u0 ,0)), is
uÖM < v '
2
(lokaal) voldoende en noodzakelijk voor (lineaire) stabiliteit.
Roache, P.J., Computational Fluid Dynamics, 1976, Hermosa, Albuquerque,
p. 42-45.
2. Gradshteyn & Ryzhik's eigen definitie van Bernoulli polynomen (9.62)
laten een vereenvoudiging toe van Gradshteyn & Ryzhik (9.531).
Gradshteyn & Ryzhik, Table of Integrals, Series and Products, 1965,
Acad. Press, New York.
3. De oplossing van het probleem van de "lange zware kabel" wordt door
Plunkett gegeven in de vorm van een asymptotische reeks naar een kleine
parameter a. Slechts de eerste term van deze reeks is correct. Dit
vindt zijn oorzaak in het feit dat hij een voorkomende functie B(z,a)
ontwikkelt naar machten van a 2 in plaats van a. Dit kan de door Pedersen
aan Plunkett's oplossing toegeschreven beperkingen verklaren.
Plunkett, R., Trans ASME, J. Engin. Indus., 89, 31-36, 1967.
Pedersen, P.T., Intern. Shipbuilding Prog., ~. 399-408, 1975.
4. Beschouw de differentiaalvergelijking x = f(y') voor y = y(x) op [0,00).
Zij y0 tussen de reële getallen a en S, - 00 ~ a,S ~ 00 , met y(O) = y0 ,
f(a) = 0, f(S) = oo. Tussen a en S is f monotoon, positief en continu.
Noem f(s) = F'(s).
Indien de vergelijking asymptotisch is te inverteren tot
À -1 À -1 "-1-1 y' (x) a À x n + • . . + a À x 0
, + bx-1 + a..1 À-1 x +
dan
y(x)
n n o o
À -1 + a À x -m+l
À -1 -m
+ O(x -m+l -m+1
is À À
= n + 0 + b log a x ... + a x n 0
À À -1 -m+1 a x + ... + a -m+lx -1
(x-+- oo),
x + A +
À + O(x -m)
À
waarbij A = y + [t f(t) - F(t) - a f n(t) -o n
- b log(I + f(t))] 13 • Cl
(x+oo),
À -af 0
(t) 0
5. De procedure van het bloksgewijs matchen, die Crighton & Leppington
voorstellen, wanneer in de termen van een asymptotische reeks naar een
kleine parameter E zowel machten van E als van log E als factor voor
komen, laat zich omschrijven als: beschouw log E van orde 1. Dit is
inderdaad vaak het geval in fysische toepassingen.
Crighton, D.G. & Leppington, F.G., Proc. R. Soc. Lond. !:_, ~. 313-339,
1973.
6. Aangezien de door Shivamoggi voorgestelde vereenvoudigde matchings
procedure slechts in een beperkt aantal gevallen toepasbaar is, en dan
ofwel te zeer voor de hand ligt om een apart formalisme aan te wijden,
ofwel pas achteraf toepasbaar blijkt, levert deze procedure geen enkel
voordeel op boven de bestaande algemenere methoden.
Shivamoggi, B.K., ZAMM., 58,354-356, 1978.
7. De interpretatie door Ffowcs Williams van Howe's werk is niet correct.
Ffowcs Williams, J.E., Aeroacoustics, Ann. Rev. Fl. Mech. 1977,
p. 454, 455.
8. Beschouw het probleem van lange geluidsgolven in een halfoneindige open
pijp, waaruit lucht stroomt met Mach getal M, 0 ~ M < 1. De parameter,
representerend de faseverschuiving van de golf na reflectie tegen het
open pijpuiteinde, wordt genoemd de "eind correctie". Dimensieloos is
deze, bij afwezigheid van stroming, gelijk aan: 0,6133, en bij een
incompressibele stroming: 0,2554. In beide gevallen is M = O.
Dit proefschrift.
9. Lezen is het herkennen van woordbeelden en niet het aaneenrijgen van
letters. Wijzigingen in de spelling moeten dan ook slechts na zeer
zorgvuldige overwegingen ingevoerd worden.
JO. Indien het voor iemand openhouden van een zich zelf sluitende deur
bedoeld is als beleefd gebaar, dient nauwkeurig in acht te worden
genomen de snelheid en de afstand tot de deur van degene, tot wie dit
gebaar is gericht. Indien hij gedwongen wordt zijn pas te versnellen,
om niet zelf onbeleefd te lijken, is het gebaar niet zo beleefd meer.
Sjoerd W. Rienstra
Eindhoven, 8 juni 1979